The singular points of a rational plane curve and its restricted tangent bundle

Journal of Algebra 532 (2019) 22–54

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Journal of Algebra www.elsevier.com/locate/jalgebra

The singular points of a rational plane curve and its restricted tangent bundle Maria-Grazia Ascenzi Department of Orthopaedic Surgery, 22-69 Rehabilitation Building, University of California, Los Angeles, CA 90095, United States of America

a r t i c l e

i n f o

Article history: Received 10 August 2018 Available online 15 May 2019 Communicated by Steven Dale Cutkosky Keywords: Minimum degree relation Rational curve Restricted tangent bundle Singular point Subline bundle Syzygy

a b s t r a c t We consider the vector bundle ϕ∗ TP 2 where ϕ denotes a generically one-to-one parametrization of a rational curve D of degree dD ≥ 2 in P 2 . We study a parameter aD that controls the splitting of ϕ∗ TP 2 as a direct sum of line bundles. We find that specific properties of D, related to its singular points, characterize aD . These properties are: multiplicity, non-collinearity of triples, and construction of a curve of degree strictly smaller than dD with multiplicity bounded by the multiplicity of D at corresponding points. © 2019 Elsevier Inc. All rights reserved.

Contents 0. Introduction . . . . . . . . . . . . . . . 1. Prior results . . . . . . . . . . . . . . . 2. Preparatory step . . . . . . . . . . . . 3. Singular points of associated curve 4. Inductive process and main result . 5. Configuration of singular points . . 6. Characterization of splitting . . . . Acknowledgments . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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E-mail address: [email protected]. https://doi.org/10.1016/j.jalgebra.2019.04.027 0021-8693/© 2019 Elsevier Inc. All rights reserved.

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0. Introduction Throughout this paper, we consider the vector bundle ϕ∗ TP 2 where ϕ denotes a generically one-to-one parametrization of a rational curve D of degree dD ≥ 2 in P 2 over an algebraically closed field k. By abuse of language, we refer to this vector bundle of rank 2 on P 1 as the tangent bundle restricted to D. Because all curves in this paper are rational curves, by abuse of language, simply “curve” without “rational” stands for “rational curve”. Because the restricted tangent bundle splits into the direct sum of two line bundles [19], we are interested in characterizing the degrees of those line bundles in terms of the singular locus of D. If the largest multiplicity of D at a singular point is “large”, i.e. equals at least dD /2 where   denotes the floor function, then we were able to provide the characterization [2,3]. The curves with a large multiplicity at one of their singular points were named “Ascenzi” curves [16]. In this paper, we complete the characterization in terms of the singular points for curves D that are not “Ascenzi”, i.e. have only multiplicity < dD /2 at their singular points. These multiplicities are called “small”. We recall that the structure of the tangent bundle restricted to a curve in P n, with n ≥ 2, has been extensively investigated since the 1980s. For example, the projective normality of a curve of positive geometric genus was found not to be a sufficient condition for the restricted tangent bundle to split [25]. Furthermore, the question of the stability and semi-stability of the restricted tangent bundle was addressed [5,21,22]. Numerous authors studied the tangent bundle restricted to rational curves [2–4,6–10,12–18,21–24, 26]. More specifically, the splitting of the tangent bundle restricted to a rational curve in P 2 was linked to the determination of an implicit equation from a parametrization of a curve in terms of moving lines [7] and to the deduction of the minimal free resolution of fat point subschemes through the cotangent bundle [15–18]. An outline of the paper is presented here. We use the notation: ϕ∗ TP 2 ∼ = OP 1 (dD + d1D ) ⊕ OP 1 (dD + d2D )

(1)

where d1D , d2D are positive integers with d1D ≤ d2D and d1D + d2D = dD . Hence, 0 < d1D ≤ dD /2 and dD /2 ≤ d2D ≤ dD − 1 where  denotes the ceiling function. Furthermore, in order to emphasize the difference in degree between the two subline bundles in (1), we use the notation ϕ∗ TP 2 ∼ = OP 1 (3dD /2 − aD ) ⊕ OP 1 (3dD /2 + aD )

(2)

where aD denotes an integer. Therefore, 0 ≤ aD < dD /2 and aD = dD /2 − d1D = d2D − dD /2 . The restricted tangent bundle is said to be balanced when the difference between the degrees of the two subline bundles in (2) is minimized; that is, aD = 0. In Section 1, we recall that the splitting of the restricted tangent bundle is determined by the relation of minimum degree among the parametric coordinates ϕ0, ϕ1 , ϕ2 of degree

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M.-G. Ascenzi / Journal of Algebra 532 (2019) 22–54

dD of ϕ (Lemma 1.2). Then, we list the relevant prior results mainly regarding curves with a large multiplicity at one of their singular points. In Section 2, we introduce our approach. We present the minimum setup necessary to explain a preparatory step and prove the simplest result about aD for a curve D with only small multiplicities at its singular points. This result provides an example of the type of results that our approach can yield. Afterwards, we contrast this result with the full complexity of our approach. Our approach is based on the largest multiplicities of D occurring at three noncollinear points (Definition 2.1 and Lemma 2.2). From them, we construct a parametrization χ = [χ0 , χ1 , χ2 ] of a curve C in P 2 of degree dC , with dC < dD . We define the three polynomials χ0 , χ1 and χ2 of same degree (Definitions 2.4 and 2.6) in such a way that the multiplicities of C at the three non-collinear points where D has the largest multiplicities are the smallest. This requirement yields a curve C with minimum degree possible under our construction. C is called minimum-degree curve associated with the parametrization ϕ of D. Proposition 2.14 states: Proposition 0.1. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points, of which 3 are non-collinear with largest multiplicity (dD /2 − 1). If the map χ is generically one-to-one, then aD = 0; otherwise, aD = 1. In this proposition, if the map χ is not generically one-to-one, then dC = 1. We see in Section 5 that if χ is not generically one-to-one and dC = 1, then we can construct a curve DM of degree 2(dD /2 − 1) such that for each singular point q ∈ DM that is not an infinitely near point, there exists a point p ∈ D such that μq DM ≤ μp D, where μr E denotes the multiplicity of E at r. In Section 2, we see that a curve D with the properties listed in Proposition 2.12 uniquely defines the minimum-degree curve C. For a curve D with only small multiplicities at its singular points, more than one curve may satisfy the definition of being a minimum-degree curve. Proposition 2.13 shows that these curves share the same minimum degree of a relation among their parametric coordinates. We determine that aD in Proposition 0.1 depends on: (i) the three largest multiplicities of D at non-collinear points, which led us to study the singular points of C (Section 3); (ii) whether the map χ = [χ0 , χ1 , χ2 ] of C is either generically one-to-one (that we call unfactorable, Definition 2.7) or factorable; and (iii) the existence of a large multiplicity of C at a singular point, unless C is a conic or a line, which led us to investigate also the possibility that C has only small multiplicities at its singular points (Section 4). In Section 4, we iterate the construction of a minimum-degree curve ˜j-times, where ˜j > 0 and ˜j depends on D. If ˜j ≥ 1, at each of (˜j − 1) iterations for 1 ≤ j ≤ ˜j − 1, we construct a curve Cj with only small multiplicities at its singular points. At the ˜j-th iteration, the curve C˜j has at least one large multiplicity at a singular point, unless C˜j is a conic or a line. Because of the potential lack of uniqueness discussed in Section 2, we ˜ obtain a tree of curves, of which any given sequence MD = {Cj }jj=1 with ˜j ≥ 1 allows us to compute d1D .

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For j = 0, ..., ˜j − 1, we let m2,j denote the smallest multiplicity among the three non-collinear points of the largest multiplicity used for the construction of Cj+1 . For j = 0, ..., ˜j, we denote (i) c j the quotient deg χ/dC (see Definition 2.7 and Remark 2.10); (ii) dCj the degree of Cj for j = 0, ..., ˜j; and (iii) m0,˜j the largest multiplicity of C˜j at a point if C˜j is not a line. We compute aD from d1D as (dD /2 − d1D ) (see explanation after Equation (2)). Our main Theorem 4.7 states: Theorem 0.2. Let D denote a curve with only small multiplicities at its singular points. ˜ Let MD = {Cj }jj=1 denote a sequence of minimum-degree curves associated with D. Then, ˜

d1D =

˜

j−1  j j   ( cr m2,j ) + cr d1C˜j j=0 r=0

(11)

r=0

and

aD

⎢ ⎥ ˜ ⎢ ˜j ⎥ j ⎢  ⎥  ⎣ ⎦ = cr dC˜ /2 − cr d1C˜ j

r=0

j

(12)

r=0

with

d1C˜j

⎧ ⎪ d − m0,˜j ⎪ ⎨ C˜j = m0,˜j ⎪ ⎪ ⎩ 0



if m0,˜j > dC˜j /2 > 0

if m0,˜j = dC˜j /2 > 0 if dC˜j = 1.

In Section 5, we discuss the properties of the singular locus of D that are relevant to Theorem 4.7. Example 5.3 analyzes: (i) the multiplicities of D at its singular points; (ii) the configuration of the singular points in terms of the non-collinearity of the singular points used in the construction of MD ; and (iii) the existence/non-existence of a curve DM of degree smaller than D satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D. In Section 6, we present five corollaries to our main Theorem 4.7. It is known that a general curve D has a balanced restricted tangent bundle and only ordinary nodes as singular points ([2], [20]). We have the following result for arbitrary curves (Corollary 6.2): Corollary 0.3. Let D be an arbitrary curve of degree dD ≥ 6 with only proper singular points and multiplicity 2 at each singular point. Let MD denote a sequence of minimumdegree curves associated to D. Then aD ≡ c ˜j /2 mod 2 with c ˜j ≥ 1. Furthermore, we list the conditions that characterize the curves with aD = dD /2 − d1D for d1D ≥ 2 (Corollaries 6.3, 6.6 and 6.7). If d1D = 2, then Corollary 6.3 states:

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Table 0 Values of multiplicity and parameters relative to Equation (11) for aD = dD /2 − 2. ˜ j

m2,0

c 1

d1C1

dC1

1

2

dD − 4

0

1

Corollary 0.4. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. Then, aD = dD /2 − 2 if and only if the case in Table 0 occurs. The curve D has three proper singular points of multiplicity 2 that are not collinear. Furthermore, there exists a curve DM of degree 5 satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM = μp D = 2. We show (Corollary 6.8): Corollary 0.5. For any pair of integers (d, d1 ) with d ≥ 6 and 2 ≤ d1 ≤ d/2, there exists a curve D of degree dD = d with only small multiplicities at its singular points and d1D = d1 (i.e. aD = dD /2 − d1 ). 1. Prior results In this section, we let D denote a plane curve of degree dD ≥ 2. Let [t, s] denote a pair of homogeneous coordinates of P 1 . In Definition 1.1 and Lemma 1.2, let ϕ : P 1 → P 2 with ϕ = [ϕ0 , ϕ1 , ϕ2 ] denote a generically one-to-one parametrization of D. Definition 1.1. Let Ai ∈ k[t, s], i = 0, 1, 2, denote homogeneous polynomials of the same degree n with 2 

Ai ϕi ≡ 0.

(3)

i=0

We say that the polynomials Ai define a relation of degree n among the parametric coordinates ϕi with i = 0, 1, 2. The length of the relation is the number of non-zero polynomials Ai , either 2 or 3. The relation is primitive if the greatest common divisor of the polynomials Ai has degree zero. A minimal relation is a relation of lowest degree. Lemma 1.2. The sheaf (ϕ∗ TP 2 )∗ ⊗ OP 1 (dD ) and the sheaf associated with the graded k[t, s]-module whose elements are the relations among the ϕi , are isomorphic. Proof. We refer to Lemma 1.2 in [3]. This is an application of the Hilbert-Burch theorem (for statement and proof, see [11]). We recall that we apply the function  F (−) = ν H 0 (P 1 , − ⊗ OP 1 (ν)) to the dualization of the short exact sequence of vector ϕ

ψ

bundles 0 → OP 1 −→ OP 1 3 (dD ) −→ ϕ∗ TP 2 → 0. We let S denote k[t, s] and S(−l) denote the graded S-module with basis in degree l. Proposition 5.13 (Chapter II, [20])

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F (ψ)

yields the exact sequence of graded S-modules 0 → S(−dD −d1D ) ⊕S(−dD −d2D ) −−−→ F (ϕ)

S 3 (−dD ) −−−→ S where d1 and d2 are defined in Equation (1), and F (ϕ) = (ϕ0 , ϕ1 , ϕ2 ). Since Ker F (ϕ) is the S-module whose elements are the relations among the ϕi and 2  Im F (ψ) = ( S(−diD ))(dD ), the lemma follows. 2 i=1

The statement of Lemma 1.2 indicates that determining aD is equivalent to determining a relation among the parametric coordinates of D of either the lowest or greatest degree, i.e. either d1D or d2D . Lemma 1.3. There exists a one-to-one correspondence between the set of points p on a curve D and the set of pairs consisting of a generically one-to-one parametrization ϕp and a relation of ϕp of length 2 and degree (dD − μp D), where μp D denotes the multiplicity of D at p. Proof. We refer to Lemma 1.3 in [3].

2

ˆD denote the maximum multiplicity of D at a point. We have: Let μ ˆD > dD /2, then d1D = dD − μ ˆD, d2D = μ ˆD and aD = μ ˆD − dD /2 . Lemma 1.4. If μ If μ ˆD = dD /2, then d1D = dD /2, d2D = dD /2 and aD = 0. Proof. We refer to Lemma 1.5 in [3]. 2 ˆD ≥ dD /2 if and only if there exists a generically one-to-one Proposition 1.5. μ parametrization ξp = [ξ0 , ξ1 , ξ2 ] of D such that a minimal relation among ξ0 , ξ1 and ξ2 has length 2. Proof. We refer to Proposition 1.6 in [3]. 2 Corollary 1.6. μp D < dD /2 for each p ∈ D if and only if, for any generically one-to-one parametrization ξ = [ξ0 , ξ1 , ξ2 ] of D, a minimal relation among ξ0 , ξ1 and ξ2 has length 3. Lemma 1.7. Let ψl with l = 1, ..., m and m ≥ 2 denote m polynomials of a given degree in k[t, s]. Let r1 , ..., rm denote m polynomials in k[t, s], not all of which are 0, that define a m relation of degree e for the ψl , i.e. l=1 rl ψl ≡ 0. Let J = {ψl : rl ≡ 0, l = 1, ..., m} and let max{deg gcd(J \ {ψl } : ψl ∈ J} denote the maximum degree of the greatest common divisors of the (m − 1)-tuples in J. Then, e ≥ max deg gcd(J \ {ψl } : ψl ∈ J) − deg gcd J. Proof. We refer to Lemma 1.6 in [2]. 2 Theorem 1.8. The following conditions are equivalent:

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(i) aD = dD /2 − 1; (ii) there exists a point p ∈ D at which D has a multiplicity equal to (dD − 1). Proof. This is Theorem 3 in [3]. Implication ((i) ⇒ (ii)) follows from Lemma 1.3. Implication ((i) ⇐ (ii)) follows from Lemma 1.4. 2 Since this paper examines the relationship between the splitting of the tangent bundle restricted to a rational plane curve D and D’s singular points, we reference one definition and two results about the singular points of D. A singular point of a curve D of degree dD > 2 can be either proper (i.e. not infinitely near) or improper (i.e. infinitely near) (Chapter V, [20]). The number of singular points counted with multiplicities (also called the virtual number of singular points) of D, here denoted #sD, is   expressed by the genus-degree formula as #sD = dD2−1 (Chapter IV, [20]). Further more, #sD = p μp (μp − 1)/2 where μp denotes the multiplicity and the sum is taken over all the singular points p of D, either proper or improper (Chapter V, [20]). 2. Preparatory step In this section, we present minimal definitions and terminology necessary to introduce a preparatory step and prove the simplest result about aD . This simplest result provides an example of the type of results that our approach can yield. We contrast this simplest result with the full complexity of our approach in the subsequent sections. Let ϕ denote a generically one-to-one parametrization of a curve D of degree dD ≥ 2. Let μ ˆD denote the maximum multiplicity of D at its points. In Sections 2 to 6, we assume that μ ˆD < dD /2. Therefore, dD ≥ 6. Furthermore, Corollary 1.6 implies that a minimal relation in Identity (3) has length 3. We start this section by possibly changing coordinates in P 2 to rewrite ϕ in a way that is convenient to compute the degree of one of its minimal relations of length 3. Definition 2.1. Let m0 be the maximum multiplicity of D at a point and let p0 be a point at which m0 = μp 0 D. Let m1 be the maximum multiplicity of D \ {p0 } and let p1 be a point at which m1 = μp 1 D. Let m2 be the maximum multiplicity in D \ {p0 , p1 } occurring at a point p2 , non-collinear with points p0 , p1 . Let m2 = μp 2 D. We refer to m0 , m1 and m2 as the largest multiplicities of D with m0 ≥ m1 ≥ m2 ≥ 1 occurring at three non-collinear points. By abuse of language, we refer to m0 , m1 and m2 as the largest non-collinear multiplicities of D. We refer to a triple of non-collinear points {p0 , p1 , p2 }, at which D has the largest multiplicities, as the non-collinear points of largest multiplicities. Lemma 2.2. Let ϕ : P 1 → P 2 denote a generically one-to-one map with ϕ(P 1 ) = D where D has only small multiplicities at its singular points. Let m0 , m1 and m2 denote the largest non-collinear multiplicities of D. Up to an automorphism of P 2 , we write ϕ = [ϕ01 ϕ02 ϕ 0 , ϕ01 ϕ12 ϕ 1 , ϕ02 ϕ12 ϕ 2 ]

(4)

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where ϕhj = gcd (ϕh , ϕj ) with hj ∈ {01, 02, 12}; and deg ϕ01 = m0 , deg ϕ02 = m1 , deg ϕ12 = m2 . By setting ϕ 0 = ϕ0 /(ϕ01 ϕ02 ), ϕ 1 = ϕ1 /(ϕ01 ϕ12 ), ϕ 2 = ϕ2 /(ϕ02 ϕ12 ), we have gcd (ϕhj , ϕlo ) = 1, gcd (ϕhj , ϕh  , ϕj  ) = 1, gcd (ϕhj , ϕq  ) = 1

(5)

where hj, lo ∈ {01, 02, 12} with hj = lo, and q = 0, 1, 2 with q = h, j in notation hj. Furthermore, gcd (ϕh  , ϕj  ) = 1

(6)

with (h, j) ∈ {(0, 1), (0, 2), (1, 2)}. Proof. Note that the triple {p0 , p1 , p2 }, at which the largest non-collinear multiplicities of D occur, is not necessarily unique. Regardless, because any three non-collinear points can be mapped into any other three non-collinear points by an automorphism of P 2, we can assume without loss of generality that p0 = [0, 0, 1], p1 = [0, 1, 0], p2 = [1, 0, 0]. Let ϕhj = gcd (ϕh , ϕj ) with hj ∈ {01, 02, 12}. Then, deg ϕ01 = m0 , deg ϕ02 = m1 , deg ϕ12 = m2 . We set ϕ 0 = ϕ0 /(ϕ01 ϕ02 ), ϕ 1 = ϕ1 /(ϕ01 ϕ12 ), ϕ 2 = ϕ2 /(ϕ02 ϕ12 ). Consequently, conditions (5) and (6) are satisfied. 2 Remark 2.3. Note that m2 = 1 can indeed occur. For instance, m2 = 1 for the sextic √ curve over k = Q( 5) in [1] that has one proper singular point of multiplicity 2 and nine improper singular points. On the other hand, if a curve has no improper singular points, then m2 > 1. Indeed, if a curve has only proper singular points, then their virtual     number equals dD2−1 (see last paragraph of Section 1). Since dD2−1 > dD for dD ≥ 6, the singular points cannot all be collinear. These considerations suggest that not only the multiplicity of D at its proper singular points should be considered to compute aD . We will see in Section 5 that the configuration of the singular points in terms of whether or not some of the triples of singular points are collinear is related to the value of aD . Next, we define a minimum-degree curve C. Definition 2.4. Let ϕ be a parametric representation of a curve D written as in (4). We define a parametric representation χ of a rational curve C associated with the map ϕ of D as χ = [ϕ01a ϕ02a ϕ 0 , ϕ01a ϕ 1 , ϕ02a ϕ 2 ],

(7)

where ϕhja is a factor of ϕhj for hj ∈ {01, 02} with deg ϕ01a = m0 − m2 and deg ϕ02a = m1 − m2 , and m0 ≥ m1 ≥ m2 .

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Remark 2.5. If mi = mj for some i and j in {0, 1, 2} with i = j, then the choice of ϕ01a and/or ϕ02a in Definition 2.4 may not be unique. We show in Proposition 2.13 that such a potential lack of uniqueness does not affect the degree of a minimal relation of χ. Note that the definition of χ depends on the differences (m0 − m2 ) and (m1 − m2 ), and on the consequent choices of factors of ϕ01 and ϕ02 . Such dependency is necessary to obtain triples of polynomials of the same degree for the definition of χ. We have: Definition 2.6. Let C denote χ(P 1 ). We call C a minimum-degree curve associated with the map ϕ of D. We let dC denote the degree of C. Note that μ ˆD ≥ μ ˆC. Next, we must address two cases: deg χ > dC and deg χ = dC . Definition 2.7. Let  denote a parametrization of a curve E with (P 1 ) = E. E is the zero-set of a prime ideal generated by a homogeneous polynomial fE in a set of homogeneous coordinates of P 2 . Then, deg fE ≤ deg . The map  factors uniquely into a map  from P 1 to P 1 of degree e = (deg /deg fE ) and a generically one-to-one map u from P 1 to P 2 with deg u = deg fE and u (P 1 ) = E. We define  as the factor map of . If deg fE < deg , then e > 1. In this case, we say that  is factorable. In contrast, if deg fE = deg , then e = 1. In this case, we say that  is unfactorable (or generically one-to-one). Depending on the specifications of ϕ, the map χ may be factorable. We show in Proposition 2.12 that the factorability of χ affects the calculation of aC . Here we look at examples of minimal degree curves D (dD = 6) over k = C. Example 2.8. Let ϕ = [t2 (t − s)2 (t − 2s)2 , t2 s2 (t − 3s)2 , s2 (t − s)2 (t − 5s)2 ], where t and s are homogeneous coordinates in P 1 . Then, ϕ01 = t2 , ϕ02 = (t − s)2 , ϕ12 = s2 , and χ = [(t −2s)2 , (t −3s)2 , (t −5s)2 ]. The curve C is the zero-set of fC = 16x2 −72xy+81y 2 − 8xz − 18yz + z 2 , where x, y, z are homogeneous coordinates in P 2 . Therefore, 2 = dC = degχ = 2. Hence, χ is unfactorable. David Eisenbud pointed out that Iϕ : (Iϕ : Iϕij ) = IC where we let Iϕ = (ϕ0 , ϕ1 , ϕ2 ), Iϕij = (ϕ01 , ϕ02 , ϕ12 ) and IC = (χ0 , χ1 , χ2 ). Since not all curves D with unfactorable χ satisfy the above equation, it would be interesting to characterize the curves for which the equation holds. Example 2.9. Let ϕ = [(t − s)2 (t − 2s)2 t2 , (t − s)2 (t − 3s)2 s2 , (t − 2s)2 (t − 3s)2 (−t2 − s2 )]. Then, ϕ01 = (t − s)2 , ϕ02 = (t − 2s)2 , ϕ12 = (t − 3s)2 , and χ = [t2 , s2 , (−t2 − s2 )]. The curve C is the zero-set of fC = x + y + z. Therefore, 1 = dC < deg χ = 2. Hence, χ is factorable. Remark 2.10. The notations “e ”, “” and “u ” in Definition 2.7 use the lower case letter and the Greek letter of the capital letter that denotes curve “E”. For instance, we denote

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“c ” the degree of the factor map of “χ” with χ(P 1 ) = C and we denote “χu ” the unfactorable map of χ. In these notations, deg χ = c dC and dC = (dD − 2m2 )/c . Now we look at the splitting of the restricted tangent bundle via χ and via χu . Remark 2.11. If χ is unfactorable, then we apply to C the notation (1) given for D. We write χ∗ TP 2 ∼ = OP 1 (dC + d1C ) ⊕ OP 1 (dC + d2C ), where dC ≥ 2 and d1C ≥ 1 with d1C denoting the degree of a minimal relation among the parametric coordinates of χ. If χ is factorable with dC > 1, then we write χu ∗ TP 2 ∼ = OP 1 (dC + d1C ) ⊕ OP 1 (dC + d2C ) (see Remark 2.10 for the notation χu ). Furthermore, a minimal relation among the parametric coordinates of χ has degree equal to c d1C . The motivation to construct a minimum-degree curve C associated with the map ϕ of D is to reduce the calculation of the degree of a minimum relation among the parametric coordinates of ϕ to a calculation of the degree of a minimum relation among the parametric coordinates of χ of degree lower than dD , indeed equal to (dD − 2m2 ). We show the following: Proposition 2.12. Let D be a curve of degree dD ≥ 6 with only small multiplicities and 2 let i=0 Ai ϕi ≡ 0 be a minimal relation of length 3. Then deg Ai ≥ m0 . Furthermore, deg Ai = m2 + c deg aui , where the polynomials aui define a relation among the unfactorable parametric coordinates χui of C. In particular, d1D = m2 + c d1C . Proof. Because D has only small multiplicities, Corollary 1.6 implies that a relation of minimal degree and length 3 exists. We write the left-hand side of Identity (3) in terms of the parametric coordinates of C. By substituting (4) into (3), we obtain A0 ϕ01 ϕ02 ϕ 0 + A1 ϕ01 ϕ12 ϕ 1 + A2 ϕ12 ϕ02 ϕ 2 ≡ 0. Conditions (5) imply that ϕ01 divides A2 . Hence, deg Ai ≥ deg ϕ01 = m0 . This proves the first claim. We now prove the second claim. For m0 ≥ m1 ≥ m2 , conditions (5) imply that ϕ12 divides A0 , ϕ02b divides A1 , and ϕ01b divides A2 . We write A0 = ϕ12 a0 , A1 = ϕ02b a1 , and A2 = ϕ01b a2 for some ai ∈ k[t, s]. Therefore, we have deg Ai = m2 + deg ai where the polynomials ai satisfy the identity 2 i=0 ai χi ≡ 0, with χi for i = 0, 1, 2 denoting the parametric coordinates of χ in (7). Remark 2.11 implies that deg Ai = m2 + c deg aui

(8)

where the polynomials aui define a relation of length 3 among the unfactorable para2 metric coordinates χui (Definition 2.7), i.e., i=0 aui χui ≡ 0. In particular, because the minimum deg Ai is achieved for the minimum deg aui , Equation (8) implies that d1D = m2 + c d1C . 2 As different curves D are considered with different values of m0 , m1 , m2 , if dD deˆD decreases to μ ˆC, then a curve C may or may not have a creases to dC faster than μ

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large multiplicity at a singular point. We will see in Proposition 2.12 that if the curve C has a large multiplicity at a singular point, then d1D will be determined by Proposition 2.12 and Lemma 1.4. If the curve C has only singular points with small multiplicities, then we will repeat the construction with C taking the role of D. The following proposition regards dC = 1. Proposition 2.13. Let D be a curve with only small multiplicities at its singular points. The following three statements are equivalent: (i) the map χ is factorable with deg χ = c ; (ii) d1C = 0; (iii) dC = 1. Moreover, each of (i) to (iii) can occur only if m0 = m1 = m2 . Proof. ((i) ⇔ (iii)) By Remark 2.10, deg χ = c dC . Hence, degχ = c is equivalent to dC = 1. ((iii) ⇔ (ii)) Because dC ≥ 2 ⇔ d1C ≥ 1 (conditions analogous to D’s in Equation (1)), the equivalence follows. Moreover, we assume by contradiction that m0 , m1 and m2 are not all equal to each other. Definition 2.4 and conditions (5) and (6) imply that ϕ01a divides χ0 and χ1 but not χ2 . Therefore, a parametric coordinate of χu (Remark 2.10) cannot be written as a linear combination of the other two parametric coordinates; i.e. dC differs from 1. This contradicts (iii). We conclude that m0 = m1 = m2 . 2 Example 2.9 provides an example for Proposition 2.13. Now we consider a class of curves with the largest possible multiplicity at some singular points. We have: Proposition 2.14. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. Furthermore, suppose that there are three non-collinear points with largest multiplicity (dD /2 − 1). If the map χ is unfactorable, then aD = 0; otherwise, aD = 1. Proof. By hypothesis, m0 = m1 = m2 = dD /2 − 1. Furthermore, Definition 2.4 gives deg χ = dD − 2m2,0 = dD − 2(dD /2 − 1). Hence, deg χ = 2 if dD is even; and deg χ = 3 if dD is odd. If the map χ is unfactorable, then deg χ = dC = 2 if dD is even, and deg χ = dC = 3 if dD is odd. In all cases, Lemma 1.4 gives d1C = 1, i.e. aC = dC /2 − d1C = 0. Proposition 2.12 implies that aD = 0. In contrast, if the map χ is factorable, then c = 2 if deg χ = 2, and c = 3 if deg χ = 3. In both cases, dC = 1. Proposition 2.13 implies that d1C = 0. Proposition 2.12 implies that d1D = dD /2 − 1. Hence, aD = dD /2 − d1D = 1. 2

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Note that the information required to determine aD in Proposition 2.14 is: (i) the three largest non-collinear multiplicities of D; and (ii) whether map χ is unfactorable or factorable. Both (i) and (ii) will play a role in the main Theorem 4.7. Proposition 2.14 provides an example where m0 = m1 = m2 , which implies that only one map χ exists, and subsequently only one minimum-degree curve C exists (Definition 2.4 and Definition 2.6). In contrast, if m0 , m1 and m2 are not all equal to each other, then more than one map χ may exist. We show: Proposition 2.15. Let mi = mj for some i and j in {0, 1, 2} with i = j. If at least two different factors of ϕ01 and/or ϕ02 are available to define ϕ01a and ϕ02a , respectively, then the product c d1C is independent of such choice(s). Proof. The choice(s) of factors to define ϕ01a and ϕ02a can affect whether c = 1 or c > 1 (i.e. the unfactorability or factorability of χ). Since c d1C = d1D − m2 (Proposition 2.12), d1C can be affected as well. The same equation shows that the product c d1C is independent of such choice(s). 2 3. Singular points of associated curve We start by presenting a geometric relation between the curve D and a minimumdegree curve C associated with ϕ of D in terms of their singular points. We use a map defined in terms of the quotients of each parametric coordinate of map ϕ of D and corresponding coordinate of map χ of C. Definition 3.1. Let β = [ϕ01b ϕ02b , ϕ01b ϕ12 , ϕ02b ϕ12 ] with ϕ01b = ϕ01 /ϕ01a , ϕ02b = ϕ02 /ϕ02a . Let B = β(P 1 ). We let dB denote the degree of B. Note that deg ϕ01b = deg ϕ02b = m2 (Definition 2.4) and dB = 2m2 . We denote #(B · C) the number of the points of intersection between B and C. Proposition 3.2. The following statements hold true:     (i) #sD = deg 2β−1 + deg 2χ−1 + deg β deg χ − 1; (ii) #sD−#sC = #sB +#(B ·C) −1 if and only if both maps β and χ are unfactorable; and (iii) #sD − #sC > #sB + b c #(B · C) − 1, where b is the degree of the factor map of β, if and only if either β or χ is factorable.

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Fig. 1. Two examples (i and ii) of neighborhood replacement.

Proof. (i) By definition of β, we have that dD = deg β + deg χ. Recall from the end of       Section 1 that #sD = dD2−1 for dD ≥ 2. Therefore, #sD = deg 2β−1 + deg 2χ−1 + deg β deg χ − 1. (ii) By Definition 2.7 and Remark 2.10, we have that deg β = b dB where b denotes the degree of the factor map of β and dB denotes the degree of B, and deg χ = c dC .     The equation in (i) gives #sD = b dB2 −1 + c dC2 −1 + b c #(B · C) − 1. Therefore, #sD − #sC = #sB + #(B · C) − 1 if and only if both maps β and χ are unfactorable. (iii) If β is factorable, then b dB > dB . If χ is factorable, then c dC > dC . Furthermore, n−1 is an increasing function of n. If either β or χ is factorable, then the equation in 2 (i) gives the inequality #sD > #sC + #sB + b c #(B · C) − 1. Viceversa, we assume by contradiction that both β and χ are unfactorable. Then, the equation in (ii) gives #sD − #sC = #sB + #(B · C) − 1 > #sB + b c #(B · C) − 1. This is impossible because b = c = 1. We conclude that either β or χ is factorable. 2     Of the terms on the right-hand side of equation #sD = deg 2β−1 + deg 2χ−1 + deg β−1 deg χ−1 and depend deg β deg χ − 1 (Proposition 3.2), the first two terms 2 2 on the factorability of β and χ, respectively. The third term deg β deg χ does not depend   on the factorability of either β or χ. If χ is factorable, then the contribution of deg 2χ−1 dC −1 because deg χ = c dC with c > 1 (Remark 2.10). If to #sD is decreased to 2 dC −1 dC > 2, then > 0. 2 The formula (ii) for unfactorable β and χ in Proposition 3.2 can be explained as follows. Because #sD = #sB + #sC + #(B · C) − 1, in order to obtain a single curve D with #sD singular points from two distinct curves B and C, we need to reduce the multiplicity at an intersection point of B · C by 1. This can be achieved by replacing a small neighborhood of an intersection point with a small neighborhood where the multiplicity of the intersection is reduced by one. Fig. 1 shows two examples where (i) the multiplicity of the intersection decreases from 1 to 0; and (ii) B and C intersect at an ordinary node of C, and the multiplicity decreases from 2 to 1.

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Definition 3.3. We say that a point p ∈ D with p = ϕ([tp , sp ]) corresponds to a point q ∈ B (or q ∈ C, respectively), if q = β([tp , sp ]) ∈ B (or q = χ([tp , sp ]) ∈ C, respectively). Remark 3.4. Note that if m0 = m1 = m2 and p ∈ {p0 , p1 , p2 }, then p corresponds to a point q ∈ B but not to any point q on C. Furthermore, a proper singular point p ∈ D, with p ∈ / {p0 , p1 , p2 }, may correspond to a proper singular point of B that is not on C. Also, a given proper singular point of D may correspond to an intersection point of B and C. Note that, by definition of B and C, the curve C is smooth at the intersection points with B \ {p0 , p1 , p2 } where p0 , p1 and p2 were the largest non-collinear points used in the construction of C. These observations explain the decrease in the number of     singular points from dD2−1 to dC2−1 due to dD > dC = (dD − 2m2 )/c . Furthermore, we have: Lemma 3.5. Let p ∈ D correspond to a point q ∈ C. If p ∈ {p0 , p1 }, where p0 and p1 are the two points of largest multiplicities on D used to construct C, then μp D > c μq C where μq C denotes the multiplicity of C at q. Otherwise, if p ∈ / {p0 , p1 }, then μp D = μq C. Proof. The condition p ∈ {p0 , p1 } requires that the largest non-collinear multiplicities m0 , m1 and m2 are not all equal to each other. If m0 > m1 , then the point p = p0 = [0, 0, 1] corresponds to the point q = q0 = [0, 0, 1] (Definition 2.4). If m1 > m2 , then the point p = p1 = [0, 1, 0] corresponds to the point q = q1 = [0, 1, 0]. Then, μpi D = mi > mi − m2 = c μqi C for i = 0, 1. Hence, μp D > c μq C. Note that only the factors of ϕ01 and ϕ02 can contribute to the multiplicity at p and q because of conditions (5). / {p0 , p1 }. Then ϕhj ([tp , sp ]) = 0 for hj ∈ {01, 02, 12} where Otherwise, let p ∈ p = ϕ([tp , sp ]) and q = χ([tp , sp ]) ∈ / {[0, 0, 1], [0, 1, 0]}. Hence, p = [xp , yp , zp ] and q = [xq , yq , zq ] with at least two corresponding coordinates different from 0. We consider the first subcase p = [xp , yp , zp ] with zp = 0. Then, deg(gcd((zp ϕ0 −xp ϕ2 , zp ϕ1 −yp ϕ2 )) = μp D and deg (gcd((zq χ0 − xq χ2 , zq χ1 − yq χ2 )) = c μq C. Because gcd(ϕ01 , ϕ02 ) = 1 and gcd(ϕ01 , ϕ12 ) = 1 (see first set of Equations (5)), we have that deg (gcd((zp ϕ0 − xp ϕ2 , zp ϕ1 − yp ϕ2 )) = deg (gcd((zq χ0 − xq χ2 , zq χ1 − yq χ2 ))/c . Hence, μp D = μq C. The proofs of the remaining two subcases xp = 0 and yp = 0 are similar to the proof of the first subcase with the coordinates of p, q, ϕ and χ switched appropriately. 2 Remark 3.6. By Lemma 3.5, we have that m1 = μp1 D > c μq1 C where p1 corresponds to q1 ; and also m1 = μp1 D ≥ μp D = μq C where p ∈ {D \p0 } and p corresponds to q. Hence, m1 ≥ μ ˆC. Because of the requirement of non-collinearity in Definition 2.1 regarding the largest multiplicities of D for the construction of C, it is m1 (not necessarily m2 ) that satisfies the above inequalities. Note that, if χ is factorable, then dC is smaller than what it would be if χ were unfactorable; the reduction in value arising from the division by c > 1. Therefore, there are fewer proper singular points of C that can correspond to proper singular points of

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D. We will see in Sections 5 and 6 how the existence of proper singular points for D affects the value of d1D . 4. Inductive process and main result In this section, we iterate the construction of a minimum-degree curve C associated with the map ϕ of D presented in Sections 2 and 3. Because a curve D with degree 3 ≤ dD < 6 has necessarily a large multiplicity at a singular point, Lemma 1.4 allows us to compute d1D . Hence, we let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. We have: Definition 4.1. Let ϕ denote a generically one-to-one map with ϕ(P 1 ) = D, where D has degree dD ≥ 6 and only small multiplicities at its singular points. We let: • MD = {Cj }j≥1 denote a “sequence of minimum-degree curves Cj associated with the map ϕ of D” or, by abuse of language, a “sequence of minimum-degree curves Cj associated with D”. We let j = 0 refer to D, including the notations χ0 = ϕ, C0 = D and mi,0 = mi for i = 0, 1, 2 (see Definition 2.1, Lemma 2.2). For j = 1, we set C1 = C, the minimum-degree curve associated with the map χ0 of C0 with χ1 (P 1 ) = C1 (Definitions 2.4 and 2.6). For j > 1, if Cj is a singular curve with only small multiplicities at its singular points, then we let χj+1 denote a map defining Cj+1 as χj+1 (P 1 ), where Cj+1 denotes a minimum-degree curve associated with map χj of Cj . • For i = 0, 1, 2 and j > 1, let mi,j denote the three largest non-collinear multiplicities of Cj used in the construction of Cj+1 ; and let c j+1 denote the degree of the factor map of χj+1 , i.e. c j+1 = deg χj /dCj+1 (Definition 2.7). • For j ≥ 1, a point qj = χj ([tj , sj ]) ∈ Cj is said to correspond to a point qj+1 ∈ Cj+1 if qj+1 = χj+1 ([tj , sj ]) (Definition 3.3). Remark 4.2. If Cj with j ≥ 0 is a singular curve with only small multiplicities at its singular points, then: (i) dCj+1 = (dCj − 2c j m2,j )/c j+1 (Remark 2.10); (ii) μqj Cj ≥ c j+1 μqj+1 Cj+1 where qj ∈ Cj corresponds to qj+1 ∈ Cj+1 and qj is either the first or the second of the three non-collinear points of largest multiplicities used in the construction of Cqj+1 ; and μqj Cj = μqj+1 Cj+1 where qj ∈ Cj corresponds to qj+1 ∈ Cj+1 and qj is not one of the points of largest non-collinear multiplicities used in the construction of Cj+1 (Lemma 3.5); (iii) m1,j ≥ m0,j+1 and m1,˜j−1 ≥ μ ˆC˜j (Remark 3.6); and (iv) m2,j ≥ m2,j+1 (Definition 4.1).

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˜

Lemma 4.3. The sequence MD is finite, i.e. there exists a ˜j ≥ 1 such that MD = {Cj }jj=1 . Furthermore, the last minimum-degree curve C˜j is either (I) a curve with at least one large multiplicity at a singular point; or (II) a conic; or (III) a line. Case (III) occurs if and only if either (i) the map χ˜j is factorable with deg χ˜j = c ; or (ii) d1C˜j = 0. Case (III) occurs only if m0,˜j−1 = m1,˜j−1 = m2,˜j−1 . Proof. As j increases from 0 to (˜j − 1), the decrease from dCj to dCj+1 is greater or equal to 2c j m2,j ((i) in Remark 4.2). Furthermore, dCj decreases faster than μ ˆCj decreases. ˜ Hence, for some j, either the curve C˜j of degree dC˜j ≥ 6 has at least one large multiplicity at a singular point (I), or the curve C˜j has degree dC˜j < 6. If 3 ≤ dC˜j < 6, then C˜j has at least one large multiplicity at a singular point (I). If dC˜j = 2, then C˜j is a conic (II). If dC˜j = 1, then C˜j is a line (III). Furthermore, Proposition 2.13 applied to C˜j yields the equivalence of (III) with either (i) or (ii), and the necessary condition m0,˜j−1 = m1,˜j−1 = m2,˜j−1 . 2 Definition 4.4. If Cj with j = 0, ..., (˜j −1) is a singular curve with only small multiplicities at its singular points, then we let ai,j with i = 0, 1, 2 denote a triple of polynomials that define a primitive relation of length 3 among the coordinates χi,j . Let cj+1 denote the degree of the map χi,j+1 and let aui,j+1 denote the polynomials defining a primitive relation among the χui,j+1 . Remark 4.5. Remark 2.10 applied to Cj with j = 0, ..., (˜j −1) implies that deg ai,j ≥ m0,j and deg ai,j = c j m2,j + cj+1 deg aui,j+1 . Iterating Equation (8) over the ˜j curves of MD yields: ˜

deg Ai =

˜

j−1  j j   ( cr m2,j ) + cr deg aui,˜j j=0 r=0

(9)

r=0

with c 0 = 1 and c r ≥ 1 for 0 < r < ˜j. If the map χr of Cr with r > 0 is unfactorable, then c r = 1; otherwise c r > 1. Moreover, (i) of Remark 4.2 gives ˜

˜

j−1  j j   dD = 2( ( cr m2,j )) + cr dC˜j . j=0 r=0

(10)

r=0

Remark 4.6. If m0,j = m1,j = m2,j for j = 0, ..., ˜j − 1, then the construction of Cj+1 excludes the choice of factor(s) within the parametric coordinates of Cj (see Definition 2.4). Hence, the sequence MD is unique. If MD is not unique (see Remark 2.5), then the iteration of Proposition 2.13 for j = 0, ..., (˜j − 1) ensures that multiple MD yield the same degree of a minimal relation among the parametric coordinates χj . We now have the main result:

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Theorem 4.7. Let D denote a curve with only small multiplicities at its singular points. ˜ Let MD = {Cj }jj=1 denote a sequence of minimum-degree curves associated with D. Then, ˜

d1D

˜

j j−1  j   = ( cr m2,j ) + cr d1C˜j j=0 r=0

(11)

r=0

and aD

⎢ ⎥ ˜ ⎢ ˜j ⎥ j ⎢  ⎥  ⎣ ⎦ = cr dC˜ /2 − cr d1C˜ j

r=0

(12)

j

r=0

with

d1C˜j

⎧ ⎪ d − m0,˜j ⎪ ⎨ C˜j = m0,˜j ⎪ ⎪ ⎩ 0



if m0,˜j > dC˜j /2 > 0

if m0,˜j = dC˜j /2 > 0 if dC˜j = 1

(case 1) (case 2) (case 3).

Proof. We compute d1D . The minimum degree of the polynomials Ai in Equation (9) is achieved by the minimum degree of aui,˜j , i.e. d1C˜j . Hence, we have proved Equation (11). To compute aD , we consider three cases. Case 1: C˜j has a large multiplicity at least at one singular point (this is case (I) in

Lemma 4.3) and m0,˜j > dC˜j /2 . Lemma 1.4 applied to C˜j implies that d1C˜j = dC˜j − ˜j−1 j ˜j m0,˜j . Therefore, Equation (11) gives d1D = j=0 ( r=0 cr m2,j ) + r=0 cr (dC˜j − m0,˜j ). ˜

˜ j j  Hence, Equation (10) gives aD = dD /2 − d1D = c d /2 − r=0 cr (dC˜j − r=0 r C˜j ˜j m0,˜j ) = r=0 cr (m0,˜j − dC˜j /2), which yields Equation (12). Case 2: Either C˜j has a large multiplicity at a singular point (this is case (I) in

Lemma 4.3) and m0,˜j = dC˜j /2 , or C˜j is a conic (this is case (II) in Lemma 4.3).

Lemma 1.4 applied to C˜j implies that d1C˜j = dC˜j /2 . Hence, Equation (11) gives

˜j−1 j ˜j d1D = j=0 ( r=0 cr m2,j ) + r=0 cr dC˜j /2 . Therefore, Equation (10) gives aD = ˜

˜

j j   r=0 cr dC˜j /2 − r=0 cr dC˜j /2 , which yields Equation (12). Case 3: The map χ˜j of C˜j is factorable with dC˜j = 1 (this is case (III) in Lemma 4.3). Lemma 4.3 implies that d1C˜j = 0. Therefore, Equation (11) gives d1D = ˜j−1 j ˜j−1 j  m2,j . Hence, aD = dD /2 − d1D = dD /2 − j=0 r=0 cr m2,j = r=0 cr

j=0 ˜j ˜ j   r=0 cr dC˜j /2 = r=0 cr /2 , which yields Equation (12). 2 Remark 4.8. We compare the values of d1D among the cases 1, 2, and 3 for fixed c r with r = 0, ..., ˜j and fixed m2,j with j = 0, ..., (˜j − 1) in Equation (11). The smallest

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value of d1D , occurring in case 3, is strictly greater than 1 because m2,0 ≥ 1 and ˜j ≥ 1. Hence, the largest value of aD , equal to (dD /2 − 1), cannot occur for curves that have only small multiplicities at their singular points. Therefore, aD = dD /2 − 1 can occur only for curves with the largest possible multiplicity (dD − 1). This result confirms our previous result in Theorem 1.8. The smallest value the largest of˜ aD occurs for 

value ˜ j j   of d1D (case 2), i.e. aD = dD /2 − d1D = aD = c d /2 − c r=0 r C˜j r=0 r dC˜j /2 . Hence, aD = 0 can occur for curves that have only small multiplicities at their singular points. Proposition 2.14 for unfactorable χ provides an example. Now we look at the balanced restricted tangent bundle. Specifically, we examine the case where aC˜j = 0 implies aD = 0. Proposition 4.9. Let D denote a curve with only small multiplicities at its singular points. Let MD denote a sequence of minimum-degree curves associated with D. The tangent bundle restricted to D is balanced if and only if the tangent bundle restricted to terminal C˜j of MD is balanced and either (i) dC˜j is even; or (ii) dC˜j is odd and the maps χj with j = 1, ..., ˜j are unfactorable. ˜

˜

j j   Proof. aD = 0 if and only if aD = c d /2 − c /2 with dC˜j > 1 d C˜j r=0 r C˜j r=0 r (Remark 4.8). For m, n ∈ N with m ≥ 1 and n > 1, we have that mn/2 = mn/2 if and only if either n is even or n is odd and m = 1. Hence, aD = 0 if and only if either (i) dC˜j is even; or (ii) dC˜j > 1 is odd and c r = 1 (i.e. χr is unfactorable) for r = 1, ..., ˜j. 2 Note that the conditions c j = 1 for j = 1, ..., ˜j in Proposition 4.9 imply by (i) of Remark 4.2 that (dCj − dCj+1 ) = 2m2,j for j = 1, ..., (˜j − 1). Therefore, (dCj − dCj+1 ) is even. Hence, if dC˜j is odd, then dD is odd as well. Furthermore, for aD to achieve its minimum value, the value of d1D in Equation (11) has to be maximum. This occurs if both d1C˜j and dCj with j = 1, ..., (˜j − 1) are maximum. Note that the maximum value of dCj occurs when both m2,j−1 is minimum and c j is minimum, i.e. c j = 1 for j = 1, ..., (˜j − 1) (see (i) of Remark 4.2). Next, we use Lemma 1.2 to construct a diagram of S-modules, with S = k[t, s], that illustrates the splitting of the tangent bundle of D. We recall that S(−l) denotes the graded S-module with basis in degree l. Remark 4.10. By means of a sequence MD consisting of ˜j curves with ˜j > 0, Theorem 4.7 allows us to compute the splitting of the tangent bundle restricted to D. We do so from specific properties of the singular points of the first (˜j − 1) curves of MD and from the splitting of the tangent bundle restricted to the last curve of MD . We show a corresponding diagram of S-modules for the case where ˜j > 1 and the maps χj with j = 1, ..., ˜j are unfactorable:

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0

S(−dD − d1D ) ⊕ S(−2dD − d1D )

F (ψ)

S 3 (−dD )

S(−dC1 − d1C1 ) ⊕ S(−2dC1 − d1C1 )

S

τ1

ρ1

0

F (ϕ)

F (ψ1 )

S 3 (−dC1 )

id F (χ1 )

S

τ2

.. . τj

.. . τ˜j

0

S(−dC˜j − d1C˜j ) ⊕ S(−2dC˜j − d1C˜j )

F (ψ˜j )

S 3 (−dC˜j )

F (χ˜j )

S.

In bold, we show the S-modules directly involved in the computation of d1D . The maps relative to the construction of C1 from D (Lemma 2.2 and Definition 2.4) are listed here: 

• • • • •

   A 0 R0 a0 r0 F (ψ) = A1 R1 , F (ψ1 ) = a1 r1 ; A2 R2 a2 r 2    ϕ01b ϕ02b 0 0 ϕ01 ϕ02 ϕ12 0 ϕ01b ϕ12 0 τ1 = , ρ1 = 0 0 0 ϕ02b ϕ12    F (ϕ) = (ϕ01 ϕ02 ϕ 0 ϕ01 ϕ12 ϕ 1 ϕ02 ϕ12 ϕ 2 ); F (χ1 ) = (ϕ01a ϕ02a ϕ 0 ϕ01a ϕ 1 ϕ02a ϕ 2 ); “id” denotes the identity map;

 ρ12 ; ρ22

with A0 = ϕ12 a0 , A1 = ϕ02b a1 , A2 = ϕ01b a2 ; deg Ai = d1D , deg Ri = dD − d1D , deg ai = d1C1 , deg ri = dC1 − d1C1 for i = 0, 1, 2 (see statement and proof of Proposition 2.12). An analogous diagram can be drawn if there is at least one χj map of Cj with j in {1, ..., ˜j} that is factorable. If so, factorable map(s) ρj and χj must be replaced by the unfactorable map(s) ρuj and χuj , respectively. We will refer to the above diagram in conjunction with Proposition 6.5. 5. Configuration of singular points In this section, we use main Theorem 4.7 to explain how the configuration of the singular points of D and the multiplicities of D at its singular points influence the computation of aD . The construction of the curves Cj in MD allows us to write the original parametrization ϕ of D as

M.-G. Ascenzi / Journal of Algebra 532 (2019) 22–54

ϕ=

˜ j−1 

(

j 

41

˜

Tr

−1

(βj+1 ))

j=0 r=0

j 

Tj −1 (χ˜j )

(13)

j=0

where Tr is the automorphism of P 2 that brings a specific triple of non-collinear points at which Cr has the largest multiplicities to [0, 0, 1], [0, 1, 0] and [1, 0, 0] (Lemma 2.2 and Definition 4.1); and βr for 1 ≤ r ≤ ˜j are defined in Definition 4.1. We substitute χu˜j (Definition 4.4) for χ˜j in Equation (13) and define

ϕM =

˜ j−1 

(

j 

j=0 r=0

˜

Tr

−1

(βj+1 ))

j 

Tj −1 (χu˜j ).

(14)

j=0

Then, we set DM = ϕM (P 1 ). Remark 5.1. Note that Equation (13) allows us to construct examples of curves D for any given value of aD . We would start from χ˜j and then continue with βj for decreasing j from ˜j to 1. In particular, Examples 2.8 and 2.9 for dD = 6 (implying ˜j = 1) were obtained from β = β1 and χ = χ1 by means of Lemma 2.2 and Definition 2.4. Specifically, we choose ϕ01a , ϕ01b , ϕ02a , ϕ02b , ϕ12 , ϕ0 , ϕ1 , ϕ2 satisfying (5) and (6), and the desired aC . Proposition 5.2. Let D denote a curve with only small multiplicities at its singular points. Let MD be a sequence of minimum-degree curves associated with D. The map χ˜j is factorable with dC˜j = 1 if and only if (i) the degree of DM , equal to (dD − c˜j + 1), is strictly smaller than dD ; and (ii) for each proper singular point q ∈ DM , there exists a point p ∈ D with μq DM ≤ μp D. Proof. (⇒) Condition (i) is satisfied because deg ϕM = deg ϕ − (c˜j − 1). Condition (ii) ˜j−1 j is satisfied because the maps ϕM and ϕ share the factor j=0 r=0 Tr −1 (βj+1 ) and (ii) of Remark 4.2 applies. (⇐) We assume by contradiction that χ˜j is unfactorable. Then, c˜j = 1. Hence, the curve DM has degree equal to dD , which contradicts condition (i). We conclude that χ˜j is factorable. Furthermore, the degree (dD − dC˜j (c˜j − 1)) computed for a factorable χ˜j cannot equal the degree of a curve satisfying (ii) for some unfactorable χ˜j because the number of singular points of this curve would be strictly smaller than #sD. 2 Example 5.3. Let D be a curve of degree 6 or 7 with only small multiplicities at its singular points. This implies that μ ˆD = 2, and so dC1 = (dD − m2,0 )/c ≤ 5 (see (i) of Remark 4.2). Thus, either C1 is smooth of degree 1 or 2, or else singular of degree 3, 4 or 5. Furthermore, 2 = μ ˆC1 ≥ dC1 /2. Hence, MD consists of just a single curve C1 . We list in Table 1 the possible values of m2,0 and c 1 as well as the values for d1C1 and d1D as computed by Equation (11). Note that the conditions (iii) of Remark 4.2 are satisfied. Note that all the values from 0 to dD /2 − 2, which equals 1, can occur for aD (see Remark 4.8).

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Table 1 Computation of aD for curves of degree 6 and 7. n

dD

m2,0

c 1

d1C1

dC1

μ ˆ C1

d1D

aD

1 2 3 4

6 6 6 6

2 2 1 1

2 1 2 1

0 1 1 2

1 2 2 4

1 1 1 2

2 3 3 3

1 0 0 0

1 2

7 7

2 2

3 1

0 1

1 3

1 2

2 3

1 0

The sextic curves numbered by n from 1 to 4 in Table 1 of Example 5.3 differ in terms of (i) the non-collinearity of the singular points at which the curves have largest multiplicities, (ii) the maximum multiplicity of C1 at a point, and (iii) either the existence or non-existence of a quintic DM such that for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D = 2 (Proposition 5.2). Furthermore, we have: • For n = 1, 2, the sextics have three non-collinear points of multiplicity 2: the existence of the quintic DM for n = 1 (Proposition 5.2) differentiates n = 1 from n = 2. These are the two cases where C1 is smooth. For n = 1, three additional singular points (that are collinear) correspond to the intersection of C1 with B1 (Proposition 3.2 and Remark 3.4). For n = 2, seven additional singular points (that lie on a conic) correspond to the intersection of C1 with B1 . See Example 2.9 for n = 1 and Example 2.8 for n = 2. • For n = 3, 4, the sextics do not have three non-collinear singular points of multiplicity 2. The sextic n = 3 has either only one proper singular point of multiplicity 2 (m0,0 = 2 and m1,0 = m2,0 = 1) or at least two proper singular points of multiplicity 2 (m0,0 = m1,0 = 2 and m2,0 = 1). No other singular point of D can correspond to a singular point of C1 ((ii) of Remark 4.2) because C1 is smooth. The sextic n = 4 has three collinear proper singular points (m2,0 = 1). Three additional singular points correspond to the singular points of the quartic C1 . • For n = 1, 2, the septics have three non-collinear points of multiplicity 2. A quintic DM (Proposition 5.2) exists for n = 1. For n = 2, the septic has at least four proper singular points: three non-collinear points of multiplicity 2 used in the construction of C1 and a fourth point that corresponds to the singular point of C1 . Note that, within Table 1: • The fact that aD = 0 for n > 1 can be viewed as a consequence of Proposition 2.14. • The fact that aD equals either 0 or 1 can be viewed as a consequence of Proposition 4.9. • aD = 1 only for n = 1. The value 1 for aD corresponds to the existence of a line serving as a minimum-degree curve.

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43

Fig. 2. Construction of Cj with j = 1, ..., ˜ j for ˜ j > 1.

However, the last statement is not necessarily true if dD > 7. Indeed, dD = 8 with m0,0 = m0,1 = m0,2 = 2, c 1 = 4 and dC1 = 1 in Equation (12) yields an example where aD = 2 (see also upcoming Corollary 6.3). Remark 5.4. The above results indicate that d1D depends on the configuration of the singular points of D. Theorem 4.7 shows that d1D also depends on m2,j for j = 0, ...(˜j−1). For instance, m2,0 = 1 if and only if one of three cases occurs: (i) there is only one proper singular point (and the remaining singular points are improper); (ii) exactly two proper singular points exist; and (iii) at least three proper singular points exist and all the proper singular points are collinear. In all three cases, dC1 = (dD − 2m2,0 )/c 1 = (dD − 2)/c 1 ((i) of Remark 4.2 with ˜j = 1 and D = C0 ). We compare a configuration for which m2,0 = 1 with a second configuration of a curve of same degree and same values for m0,0 , m1,0 but m2,0 > 1. We assume that all points p have multiplicity smaller than m2,0 except for p0 and p1 . If the degree of χ1 for the first configuration equals the degree of χ1 for the second configuration, then m0,1 is higher in the first configuration. Note that this higher value of m0,1 influences the construction of C2 , including its degree. Equation (11) gives the value of d1D in terms of m2,j and c j for j = 0, .., .(˜j − 1), as well as m0,˜j if dC˜j > 1. We have c j = deg χj /dCj (Definition 4.1). Furthermore, Cj depends on the largest non-collinear multiplicities mi,j for i = 0, ..., 2 and j = 0, ..., (˜j−1), and m0,˜j . Therefore, the value of d1D depends on (i) the triples of largest non-collinear multiplicities used to construct the curves in MD up to the penultimate curve, (ii) the maximum multiplicity of the last curve at a point, and (iii) the existence/non-existence of a curve DM of degree smaller than dD such that for each proper point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D. Fig. 2 illustrates the construction of the curves Cj with j = 0, ..., ˜j for ˜j > 1. In Fig. 2, an ordinary node represents a singular point at which Cj has a small multiplicity or a

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M.-G. Ascenzi / Journal of Algebra 532 (2019) 22–54

Fig. 3. Two possible configurations of triples T1 and T2 .

smooth point for either m1,j or m2,j . A triple node represents a singular point at which C˜j has a large multiplicity. C˜j can be a curve with at least one large multiplicity at a singular point, a conic or a line (Lemma 4.3). Fig. 3 shows two possible configurations of triples (T1 and T2 ; each notation is placed inside the relevant spanned triangle) for the construction of Cj and of Cj+1 for some j = 0, ..., (˜j − 2) when mi,j = mh,j for some i and h = 0, 1, 2. Additional considerations on multiplicities and configuration of the singular points of D appear in Section 6. Remark 5.5. From Definition 4.1 and Lemma 4.3, consider the set up C0 = D and ˜ MD = {Cj }jj=1 with ˜j ≥ 2. Suppose that one of the three cases in Remark 5.4 occurs for Cjo with jo in {0, ..., (˜j − 1)}: (i) only one proper singular point exists; (ii) exactly two proper singular points exist; and (iii) all (at least three) the proper singular points are collinear. Then, m2,j = 1 for j = jo , ..., (˜j − 1). In particular, if jo = 0, then C0 = D and m2,j = 1 for j = 0, ..., (˜j − 1). The configurations in Fig. 3 can occur, depending on the multiplicities of the singular points. 6. Characterization of splitting In this section, we present five corollaries to Theorem 4.7. We start by presenting a lemma that links the maximum value of ˜j in MD , denoted ˜jmax , with d1D . Lemma 6.1. Let D be a curve of degree dD ≥ 6 with only small multiplicities at its singular points. Let MD be a sequence of minimum-degree curves associated with D. Then, in Equation (11): (i) m2,˜j−1 , c ˜j and d1C˜j cannot all equal 1. (ii) If d1D = 2, then ˜jmax = 1. Specifically, m2,0 = 2, d1C˜j = 0 and c 1 = dD − 2m2,0 > 1. (iii) Let d1D > 2. If dD is even, then ˜jmax = d1D − 2. If dD is odd, then d1D > 3 and ˜jmax = d1D − 3.

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45

Proof. The right-hand side of Equation (11) consists of the sum of (˜j + 1) terms of which the first ˜j terms are positive integers and the last (˜j + 1) term is a non-negative integer. The first term equals m2,0 because χ0 is unfactorable, i.e. c 0 = 1, by hypothesis. (i) We suppose by contradiction that m2,˜j−1 = c ˜j = d1C˜j = 1. By Lemma 4.3, the condition d1C˜j = 1 implies that either C˜j is a conic or there exists a singular point on C˜j of multiplicity equal to (dC˜j − 1) (Theorem 1.8 applied to C˜j ). The conditions m2,˜j−1 = c ˜j = 1 and (i) of Remark 4.2 give dC˜j−1 = dC˜j + 2. If C˜j is a conic, then dC˜j−1 = dC˜j + 2 = 4. Hence, C˜j−1 has a large multiplicity at a singular point. This is a contradiction because C˜j−1 can only have small multiplicities at its singular points. We conclude that C˜j is not a conic. If C˜j has a singular point of multiplicity (dC˜j −1), then (ii) of Remark 4.2 implies that also C˜j−1 has a singular point of multiplicity (dC˜j −1). Because

C˜j−1 has only small multiplicities at its singular points, we have dC˜j − 1 < dC˜j−1 /2 with dC˜j−1 = dC˜j + 2. Hence, dC˜j−1 < 6. Therefore, dC˜j−1 has a large multiplicity at a singular point. This is a contradiction. Therefore, C˜j does not have a singular point of multiplicity (dC˜j − 1). We conclude that m2,˜j−1 , c ˜j and d1C˜j cannot all equal 1. (ii) If d1D = 2, then (i) excludes m2,0 = d1C˜j = c 1 = 1. Hence, m2,0 = 2, d1C˜j = 0 and ˜jmax = 1. Item (i) of Remark 4.2 and Lemma 4.3 give c 1 = dD − 2m2,0 > 1. (iii) Let d1D > 2. If d1C˜j = 0, then item (III) of Lemma 4.3 implies that m2,˜j−1 > 1. ˜

Furthermore, the sequence {m2,j }j−1 j=0 is non-increasing ((iv) of Remark 4.2). Therefore, in order to compute the maximum value of ˜j we need to consider the case where each of the (˜j + 1) terms of Equation (11) equals 1. Hence, m2,˜j−1 = c ˜j = d1C˜j = 1, which (i) excludes. Consequently, ˜jmax ≤ d1D − 2. Suppose that each of the first ˜j terms of Equation (11) equals 1 and that the last term of Equation (11) equals 2. Then, c ˜j d1C ˜j = 2, c j = 1 for j = 1, ..., (˜j − 1) and m2,j = 1

for j = 0, ..., (˜j −1). Let c ˜j = 1 and d1C˜ = 2. Because d1C˜ ≤ dC˜ /2 , we have dC˜ ≥ 4. j

j

j

j

Lemma 4.3 and Theorem 1.8 applied to C˜j imply that C˜j has a large multiplicity (dC˜j −2) at a singular point. Then (ii) of Remark 4.2 implies that also C˜j−1 has a singular point of multiplicity (dC˜j − 2). Because C˜j−1 has only small multiplicities at its singular points,

we have dC˜j − 2 < dC˜j−1 /2 with dC˜j−1 = dC˜j + 2 ((i) of Remark 4.2). Then dC˜j = 4. Equation (10) gives dD = 2˜j + c ˜j dC˜j . Hence, dD is even. We have shown that if dD > 2, then dD is even and ˜jmax = d1D − 2.

Note that if c ˜j = 2 and d1C˜j = 1, then (i) of Remark 4.2 gives dC˜j = (dC˜j−1 − 2)/2. Equation (10) gives dD = 2˜j + c ˜j d1C ˜j = 2˜j + 2. Hence, dC˜j and dD are both even. If dD is odd, no other set of values for the parameters in Equation (11) yields ˜jmax = d1D − 2. Hence, if dD is odd, then d1D > 3. We show that if dD is odd, then ˜jmax ≤ d1D − 3. Suppose that each of the first ˜j terms of Equation (11) equals 1 and that the last term equals 3. Then, c ˜j d1C ˜j = 3, c j = 1 for j = 1, ..., (˜j − 1) and m2,j = 1 for j = 0, ..., (˜j − 1). Equation (11) gives d1D = ˜jmax + c ˜j d1C˜j = ˜jmax + 3. Therefore, ˜jmax = d1D − 3. We let c ˜j = 3 and d1C˜j = 1. Equation (10) gives dD = 2˜j + c ˜j dC ˜j = 2˜j + 3 dC ˜j . Hence, dC ˜j is odd.

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Lemma 4.3 implies that C˜j has a large multiplicity at a singular point. Then, the large multiplicity of C˜j equals (dC˜j − 1) (Theorem 1.8 applied to C˜j ). Item (ii) of Remark 4.2 implies that also C˜j−1 has multiplicity (dC˜j − 1) at a singular point. Because C˜j−1

has only small multiplicities at its singular points, we have dC˜j − 1 < dC˜j−1 /2 with dC˜j−1 = 3dC˜j + 2 ((i) of Remark 4.2). This inequality is satisfied by any dC˜j ≥ 3. 2 The first corollary of this section (Corollary 6.2 below) refers to a class of curves with multiplicity 2 (the smallest possible) at their singular points. It is known that a general curve D has a balanced restricted tangent bundle (see [2]) and only ordinary nodes as singular points (Proposition 2.7, Chapter IV, [20]). We ask whether the tangent bundle restricted to an arbitrary (non-general) curve D with all proper singular points at which D has multiplicity 2, is balanced. Corollary 6.2. Let D be an arbitrary curve of degree dD ≥ 6 with only proper singular points and multiplicity 2 at each singular point. Let MD denote a sequence of minimumdegree curves associated to D. Then aD ≡ c ˜j /2 mod 2 with c ˜j ≥ 1. Proof. The proper singular points at which D has multiplicity 2 are too numerous to be   collinear, since dD2−1 > dD for dD ≥ 6. Hence, m0,0 = m1,0 = m2,0 = 2. Then, (ii) of Remark 4.2 implies that the curves Cj with j = 0, ..., ˜j have only proper singular points of multiplicity 2 unless C˜j is a conic or a line (Lemma 4.3). If C˜j is singular, then C˜j has a large multiplicity at a singular point. Hence, μ ˆC˜j = 2 implies that 3 ≤ dC˜j ≤ 5. Therefore, we have three cases: (i) c j = 1 for j = 1, ..., ˜j with 3 ≤ dC˜j ≤ 5; (ii) c j = 1 for j = 1, ..., (˜j − 1) and c ˜j ≥ 1 with dC˜j = 2; and (iii) c j = 1 for j = 1, ..., (˜j − 1) and c ˜j > 1 with dC˜j = 1. For case (i), we have that dC˜j = dD − 4˜j, i.e. dD ≡ dC˜j mod 4, with 3 ≤ dC˜j ≤ 5. Lemma 1.4 applied to C˜j gives aC˜j = 0. Therefore, by Proposition 4.9, aD = 0. For case (ii), Equation (10) gives dD = 4˜j + 2c ˜j , i.e. dD ≡ 2c ˜j mod 4. Lemma 1.4 applied to dC˜j = 2 gives d1C˜j = 1. Equation (11) gives d1D = 2˜j + c ˜j . Hence, aD =   dD /2 − d1D = (4˜j + 2c ˜j )/2 − (2˜j + c ˜j ) = 0. For case (iii), Equation (10) gives dD = 4˜j + c ˜j , i.e. dD ≡ c ˜j mod 4. Lemma 4.3 applied to a factorable χ˜j with dC˜j = 1 implies that d1C˜j = 0. Equation (11) gives   d1D = 2˜j. Hence, aD = dD /2 − d1D = (4˜j + c ˜j )/2 − 2˜j = c ˜j /2 > 1. Therefore,    aD ≡ c ˜j /2 mod 2 and aD > 0. 2 Previously (Theorem 4 in [3]), we saw that the largest possible value of aD for curves with only small multiplicities at their singular points was (dD /2−2). Indeed, the larger value aD = dD /2 − 1 can only occur for curves with at least one large multiplicity at their singular points (Theorem 1.8 and Remark 4.8). Recall that we can characterize the curves for which aD = dD /2 − 2, by proceeding as follows (as in [3]). Starting from a relation matrix of the primitive relations of degree 2 and degree (dD − 2) that did not explicitly relate to the multiplicities of D at its singular points, we applied the

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47

Table 2 Values of multiplicity and parameters relative to Equation (11) for aD = dD /2 − 2. ˜ j

m2,0

c 1

d1C1

dC1

1

2

dD − 4

0

1

Hilbert-Burch Theorem twice to the first and second minimal relations of degrees 2 and 1, respectively. This allowed us to construct a smooth rational curve G of degree equal to dD lying on a smooth quadric surface Q, and projecting onto D from a point outside Q. Now, we provide a characterization of the case aD = dD /2 − 2 based on a different approach that is applicable to aD ≤ dD /2−2. This new approach is based on a minimal relation of a parametrization of D that relates to the multiplicities of the singular points of D. This approach can be viewed as an application of the Hilbert-Burch Theorem to the minimal relations of the minimum-degree curve Cj with j = 0, ..., (h − 1) with h = ˜j (Remark 4.10). We show here: Corollary 6.3. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. Let MD be a sequence of minimum-degree curves associated with D. Then, aD = dD /2 − 2 if and only if the case in Table 2 occurs. The curve D has three proper singular points of multiplicity 2 that are not collinear. Furthermore, there exists a curve DM of degree 5 satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM = μp D = 2. Proof. (⇒) If aD = dD /2−2, then d1D = 2. Item (ii) of Lemma 6.1 implies that ˜j = 1, m2,0 = 2, c 1 = dD − 4 and d1C1 = 0. Case (III) of Lemma 4.3 implies that dC1 = 1 and 2 = m2,0 = m1,0 = m0,0 . Proposition 5.2 implies that there exists a curve DM of degree 5 such that for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM = μp D = 2. (⇐) Clearly, the values listed in Table 2 satisfy Equation (11) for d1D = 2 and ˜j = 1. 2 Example 2.9 provides a curve D with the smallest degree dD = 6 over k = C and aD = dD /2 − 2 (as in Corollary 6.3). This curve has multiplicity 2 at 10 proper singular points. We have that χ1 = [t2 , s2 , −t2 − s2 ]. An implicit equation of C1 is given by x + y + z = 0. Equation (11) gives d1D = m2,0 + 2d1C1 = 2. Hence aD = 1. For this example, and more in general for curves D with dD ≥ 6, only small multiplicities at their singular points and aD = dD /2 − 2, we link the characterization of Corollary 6.3 with the characterization of the Theorem 4 in [3] mentioned above. Definition 6.4. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. If m0,0 = m1,0 = m2,0 = 2, then we write the polynomials ϕhj with

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hj ∈ {01, 02, 12} (defined in (4) and satisfying Equations (5)) as ιhj t2 + κhj ts + λhj s2 . We define the matrix  ΦD =

ι12 κ12 λ12

ι02 κ02 λ02

ι01 κ01 λ01



as the coefficient matrix of the polynomials ϕhj . Proposition 6.5. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points with aD = dD /2 − 2. Let MD be a sequence of minimum-degree curves associated with D. Let Q denote the smooth quadric surface in P 3 constructed in Theorem 4 of [3] that contains a smooth curve projecting onto D from a point not on Q. The quadric Q also contains the line C1 if and only if the first or third row of the coefficient matrix ΦD provides a relation of degree 0 for C1 . Proof. Equations (5) ensure that the matrix ΦD provides an automorphism of P 2 that transforms the homogeneous coordinates [x, y, z] into the new homogeneous coordinates [X, Y, Z]. We consider the new parametrization ψ of D obtained by composing ϕ with this automorphism. Corollary 6.3 implies m0,0 = m1,0 = m2,0 = 2. Corollary 1.6 implies that the minimal relation has length 3. Therefore, we can write ψ in terms of the minors of the relation matrix   t2 ts s2 R0 R1 R2 as ψ = [s(tR2 − sR1 ), −t2 R2 + s2 R0 , t(tR1 − sR0 )], as done in the proof of Theorem 4 in [3]. Then, ψ = [sη0 , −tη0 − sη1 , tη1 ] with η0 = tR2 − sR1 and η1 = tR1 − sR0 . Following the steps of the proof of Theorem 4 in [3], we construct the parametrization X = sη0 , Y = −tη0 , Z = tη1 , W = −sη1 of the curve G in P 3 . The curve G is smooth and lies on the smooth quadric Q defined by the equation XZ − Y W = 0. Furthermore, G projects onto D from the point [0, −1, 0, 1] ∈ / Q. Then, we write the parametrization γ of the line C1 = [ϕ0 , ϕ1 , ϕ2 ] (Corollary 6.3) in the new coordinates [X, Y, Z]. We have that γ = [γ0 , γ1 , γ2 ] with γ0 = ι12 ϕ0 + ι02 ϕ1 + ι01 ϕ2 , γ1 = κ12 ϕ0 + κ02 ϕ1 + κ01 ϕ2 , γ2 = λ12 ϕ0 + λ02 ϕ1 + λ01 ϕ2 . The line [γ0 , γ1 , γ2 , 0] ∈ P 3 lies on Q if and only if either γ0 ≡ 0 or γ2 ≡ 0, i.e. either the first row or the third row of ΦD defines a relation of degree 0 for ϕ0 , ϕ1 , ϕ2 (Proposition 2.13). For Example 2.9, Proposition 6.5 implies that γ = [0, −4t −2s, 8t +3s]. Hence, C1 ⊂ Q. If we replace ϕ2  = −t2 − s2 of Example 2.9 with ϕ2  = −t2 − 2s2 , then we obtain the parametrization of a new curve D. For this curve, γ = [−s, −4t, 8t + 2s]. Hence, C1 ⊂ Q. 2 In [8], Theorem 3.1 extends the characterization of aD = dD /2 − 2 of Theorem 4 in [3] to lesser values of aD . Specifically, if D has only small multiplicities at its singular

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49

Table 3 Possible values of parameters relative to Equation (11) for aD = dD /2 − 3. n

˜ j

m2,0

c 1

d1C˜j

dD

dC˜j

1 2 3 4

1 1 1 1

3 2 1 1

dD − 6 1 1 2

0 1 2 1

≥8 6, 7, 8 6 ≥ 6 even

1 dD − 4 dD − 2 (dD − 2)/2

points, the proof of Theorem 3.1 in [8] constructs a smooth rational curve G of degree equal to dD that lies on a rational normal surface R in P h+1 and that projects onto D from (h − 1) points not on R where 3 ≤ h ≤ dD /2 and aD = dD /2 − h. Differently, we provide here a list of necessary and sufficient conditions, including conditions on singular points, for a curve D to have a given aD between 0 and dD /2. We start with the next-largest case aD = dD /2 − 3 in Corollary 6.6. Examples are provided in Table 1 of Example 5.3 for dD = 6, 7 and aD = dD /2 − 3 = 0. We show: Corollary 6.6. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. Let MD be a sequence of minimum-degree curves associated with D. Then aD = dD /2 − 3 if and only if one of the cases (n = 1, .., 4) in Table 3 occurs. Furthermore, there exists a curve DM of degree 7 for n = 1 satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D. Proof. (⇒) If aD = dD /2 − 3, then d1D = 3. Remark 4.5 implies that d1D = 3 ≥ μ ˆD. Definition 4.1 and (iii) of Lemma 6.1 imply that ˜j = 1 in MD . Equation (11) for ˜j = 1 yields the 3 = d1D = m2,0 + c 1 d1C1 with 1 ≤ m2,0 ≤ 3 and 0 ≤ c 1 d1C1 ≤ 2. All the possible values of the parameters relative to Equation (11) are the ones listed in Table 3. In fact, first we focused on the possible values of m2,0 . We recall that d1C˜j = 0 implies that c ˜j > 1 and m0,˜j−1 = m1,˜j−1 = m2,˜j−1 > 1 (Lemma 4.3). Then, we focused on the possible values of c 1 . If greater than 1, then the value of c˜j for n = 1 follows from the condition dC˜j = 1, or equivalently d1C˜j = 0 (Lemma 4.3). In fact, (i) of Remark 4.2 gives c 1 = dD − 2m2,0 = dD − 6 for n = 1. Because the condition dC˜j = 1 restricts dD through the value of c˜j , we compute the ranges of dD and dCj for 1 ≤ j ≤ ˜j and dC˜j = 1. • For n = 1, we have 3 = m2,0 < dD /2. Therefore, dD ≥ 8. Here m0,0 = m1,0 = m2,0 = 3 ((III) of Lemma 4.3). • For n = 2, (i) of Remark 4.2 gives dC1 = (dD − 2m2,0 )/c 1 = dD − 4. The condition d1C1 = 1 implies that either C1 is a conic (hence dD = 6) or there exists a point q ∈ C1 at which C1 has a large multiplicity μq C1 = dC1 − d1C1 = dD − 5 (Lemma 4.3 and Lemma 1.4 applied to C1 ). This second option implies that there exists a point p ∈ D at which D has a small multiplicity with dD /2 > μp D ≥ μq C1 = dD − 5 ((ii) of Remark 4.2). Therefore, dD = 6, 7, 8.

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• For n = 3, we have dC1 = (dD − 2m2,0 )/c 1 = dD − 2. The condition d1C1 = 2 implies that there exist a point q ∈ C1 at which C1 has a large multiplicity μq C1 = dC1 − d1C1 = dD − 4 (Lemma 4.3 and Lemma 1.4 applied to C1 ). Therefore, there exists a point p ∈ D at which D has a small multiplicity with dD /2 > μp D ≥ μq C1 = dD −4 ((ii) of Remark 4.2). Hence, dD = 6. • For n = 4, we have dC1 = (dD − 2m2,0 )/c 1 = (dD − 2)/2. Therefore, dD is even. The condition d1C1 = 1 implies that either C1 is a conic (i.e. dD = 6) or there exists a point q ∈ C1 at which C1 has a large multiplicity μq C1 = (dD − 4)/2 (Lemma 4.3). This second option implies that there exists a point p ∈ D at which D has a small multiplicity with dD /2 > μp D ≥ μq C1 = (dD − 4)/2 ((ii) of Remark 4.2). The inequality is satisfied by any dD . (⇐) Clearly, the values for m2,j−1 , c j and d1C˜j for the cases listed in Table 3 satisfy Equation (11) with d1D = 3. We saw in the first part of the proof that the values of dD and dCj with j = 1, ..., ˜j are derived from m2,j with j = 0, ..., (˜j − 1) and c j with j = 1, ..., ˜j and d1C˜j . We apply Proposition 5.2 to d1C1 = 0 for n = 1. Therefore, there exists a curve DM of degree 7 for n = 1 satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D. 2 To yield aD = dD /2 − 3, Corollary 6.6 shows that d1C˜j compensates for the values of m2,j with j = 0, ..., (˜j − 1) and c j with j = 1, ..., ˜j. Note that n = 2, 3, 4 in Table 3 for dD = 6 correspond to n = 2, 4, 3 respectively, in Table 1 of Example 5.3. We have already explained these cases in terms of the non-existence of three non-collinear singular points of maximum multiplicity and the existence of a ˜j-th minimum-degree curve with ˜j = 1 of degree dC2 = 2, 3. Furthermore, n = 2 in Table 3 for dD = 7 correspond to n = 2 in Table 1 of Example 5.3 where m2,0 = 2 is coupled with the existence of a ˜j-th minimum-degree curve with ˜j = 1 of degree (dD − 4), and m2,0 = 1 is coupled with the existence of a ˜j-th minimum-degree curve with ˜j = 1 of degree (dD − 2). For dD ≥ 8, we have n = 1, 4 in Table 3. Specifically, if three non-collinear singular points with multiplicity 3 exist, then the ˜j-th minimum-degree curve with ˜j = 1 is a line. Furthermore, if m2,0 = 1, then there exist at most two proper singular points for D. We now consider the curves D with smaller values of aD . Corollary 6.7. Let D denote a curve of degree dD ≥ 6 with only small multiplicities at its singular points. Let MD be a sequence of minimum-degree curves associated with D. Then aD = dD /2 − d1D with 3 < d1D ≤ dD /2 if and only if the parameters m2,l with l = 0, ..., (˜j − 1) and c l with l = 1, ..., ˜j and d1C˜j (defined in Section 4 and satisfying the conditions of Remark 4.2, Lemma 4.3 and Lemma 6.1) are solutions of one of the ˜j l−1 j l equations d1D = j=0 ( r=0 cr m2,j ) + r=0 cr d1Cl for l = 1, ..., ˜j where ˜j ≤ d1D − 2 if dD is even and ˜j ≤ d1D − 3 if dD is odd. Furthermore, if d1C˜j = 0, then there exists

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51

a curve DM of degree (dD − c ˜j + 1) satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D. Proof. (⇒) If aD = dD /2 − d1D , then Remark 4.5 implies that d1D ≥ μ ˆD, and (iii) of Lemma 6.1 gives 1 ≤ ˜j ≤ d1D − 2 if dD is even and 1 ≤ ˜j ≤ d1D − 3 if dD is odd. l−1 j l Therefore, Equation (11) yields the ˜j equations d1D = j=0 ( r=0 cr m2,j ) + r=0 cr d1Cl for l = 1, ..., ˜j for 1 ≤ ˜j ≤ d1D − 2 if dD is even and 1 ≤ ˜j ≤ d1D − 3 if dD is odd. The values of the parameters m2,l with l = 0, ..., (˜j − 1), c l with l = 1, ..., ˜j and d1C˜j are restricted by the conditions of Remark 4.2 and Lemma 4.3. (⇐) Clearly, the values of the parameters m2,l with l = 0, ..., (˜j − 1) and c l with l−1 j l = 1, ..., ˜j and d1C˜j , that satisfy one of the ˜j equations d1D = j=0 ( r=0 cr m2,j ) + l cr d1Cl for l = 1, ..., ˜j with 1 ≤ ˜j ≤ d1D − 2 if dD is even and 1 ≤ ˜j ≤ d1D − 3 if r=0

dD is odd, satisfy Equation (11). We apply Proposition 5.2 to the cases where d1C˜j = 0. Then, there exists a curve DM of degree (dD − c ˜j + 1) satisfying the following property: for each proper singular point q ∈ DM there exists a point p ∈ D with μq DM ≤ μp D. 2 To show the pattern of the sets of values obtained with Corollary 6.7 as ˜j increases, we set forth Table 4 for d1D > 6. The sets of values place restrictions on the value of dD (as they did in Corollaries 6.3 and 6.6). In the table, w = 2 if dD is even and w = 3 if dD is odd ((iii) of Lemma 6.1). As ˜j increases from 1 to ˜jmax , the value of m2,0 decreases and the value of d1C˜j increases, both within ranges of decreasing width. Furthermore, ˜

the sequence {m2,j }j−1 j=0 is non-increasing. Furthermore, we show: Corollary 6.8. For any pair of integers (d, d1 ) with d ≥ 6 and 2 ≤ d1 ≤ d/2, there exists a curve D of degree dD = d with only small multiplicities at its singular points and d1D = d1 (i.e. aD = dD /2 − d1 ). Proof. We use the notations (4) to define a parametrization for D. If 2 ≤ d1 < d/2, then we let ϕ01 = (t − 2s)d1 , ϕ02 = (t − 3s)d1 , ϕ12 = (t − 5s)d1 ; and ϕ 0 = td−2d1 , ϕ 1 = sd−2d1 , ϕ 2 = −td−2d1 − sd−2d1 . Then, dD = d. The points [0, 0, 1], [0, 1, 0] and [1, 0, 0] of multiplicity d1 , are the singular points of D of largest multiplicity. Hence, m0,0 = m1,0 = m2,0 = d1 . Furthermore, ˜j = 1 and C1 = [t, s, −t − s]. Lemma 4.3 gives d1C1 = 0. Item (i) of Remark 4.2 yields c 1 = dD − 2d1D . Equation (11) gives d1D = m2,0 + c 1 d1C1 = d1 i.e. aD = dD /2 − d1 . If d1 = d/2, then we let ϕ01 = (t−2s)(d1 −1) , ϕ02 = (t−3s)(d1 −1) , ϕ12 = (t−5s)(d1 −1) ; and ϕ 0 = t(d−2d1 +2) , ϕ 1 = s(d−2d1 +2) , ϕ 2 = (t + 2s)(d−2d1 +1) (t + 3s). Thus, dD = d. The points [0, 0, 1], [0, 1, 0] and [1, 0, 0] of multiplicity (d1 − 1), are the singular points of D of largest multiplicity. Proposition 2.14 implies that ˜j = 1 and c 1 = 1. The curve C1 is a conic if dD is even and a cubic if dD is odd. In either case, Lemma 1.4 applied

52

Table 4 Possible values of parameters in Equation (11) for d1D > 6. m2,0

c 1

m2,1

c 2

m2,2

c 3

m2,3

c 4

...

m2,˜j−1

c ˜j

d1C˜j

1 1 1 1 1

d1D d1D d1D d1D d1D . . . 1 d1D d1D d1D d1D d1D d1D d1D d1D . . . 1 d1D d1D d1D . . . 1 . . . 1 1

dD − 2d1D 1 2 1 1 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 . . . 1 . . . 1 1

− − − − −

− − − − −

− − − − −

− − − − −

− − − − −

− − − − −

− − − − −

− − − − −

− − 2 3 2 1 3 2 2 1 . . . 1 2 1 1 . . . 1 . . . 1 1

− − dD − 2d1D dD − 2d1D 1 2 1 2 1 1 . . . 1 1 1 1 . . . 1 . . . 1 1

− − − − − − − − − −

− − − − − − − − − −

− − − − − − − − − −

− − − − − − − − − −

... ... ... ... ... . . . ... ... ... ... ... ... ... ... ...

− − − − − − − − − −

− − − − − − − − − −

− − 2 2 1 . . . 1 . . . 1 1

− − dD − 2d1D 1 2 . . . 1 . . . 1 1

− − − − −

− − − − −

− − − − −

− − − − −

− −

− −

− −

− −

− 1 1

− 1 1

− 1 1

− w 1

0 1 1 2 3 . . . d1D − 1 0 0 1 1 1 1 2 3 . . . d1D − 2 0 1 1 . . . d1D − 3 . . . 1 w

1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 . . . 3 . . . d1D − w d1D − w

−1 −2 −2 −3

−2 −3 −3 −3 −4 −4 −4 −4

−4 −4 −4

... ... ... ... ... . . . ... . . . ... ...

M.-G. Ascenzi / Journal of Algebra 532 (2019) 22–54

˜ j

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to C1 gives d1C1 = 1. Equation (11) gives d1D = m2,0 + c 1 d1C1 = d1 = dD /2 i.e. aD = dD /2 − d1 = 0. 2 Remark 6.9. For any pair of integers (d, d1 ) with d ≥ 6 and 2 ≤ d1 ≤ d/2, Remark 5.1 gives a method to construct examples of curves D with only small multiplicities at its singular points with dD = d, d1D = d1D and ˜j ≥ 1. We applied this method with ˜j = 1 in the proof of Corollary 6.8. The condition ˜j > 1 may require the use of minimum-degree curves with improper singular points (Remark 5.5, Table 4). In conclusion, we have shown that if a curve D of dD ≥ 6 has only small multiplicities at its singular points, then aD = dD /2 − d1D with d1D ≥ 2. This d1D equals ˜j ˜j−1 j   j=0 ( r=0 cr m2,j ) + r=0 cr d1C˜j (Equation (11)). Hence, d1D ≥ m2,0 ≥ 1, i.e. a lower bound for d1D is provided by m2,0 . We have demonstrated that the value of d1D depends on geometric properties of D related to its singular points available for the construction of the minimum-degree curves in MD . The properties regard multiplicity, non-collinearity of triples, and construction of a curve of degree strictly smaller than dD with multiplicity bounded by the multiplicity of D at corresponding points. Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Thanks to David Eisenbud for his openness to resume discussion on this subject thirty-two years later. T hanks to Eric Primozic and John Zhang for assisting with manuscript preparation. References [1] E. Artal Bartolo, J. Carmona Ruber, J.I. Cogolludo Agustín, On sextic curves with big Milnor number, in: A. Libgober, M. Tibăr (Eds.), Trends in Singularities, in: Trends in Mathematics, Birkhäuser Verlag, Basel, Switzerland, 2002, pp. 1–29. [2] M.-G. Ascenzi, The Restricted Tangent Bundle, PhD Thesis, Brandeis University, 1985. [3] M.-G. Ascenzi, The restricted tangent bundle of a rational curve in P 2 , Comm. Algebra 16 (1988) 2193–2208, https://doi.org/10.1080/00927878808823687. [4] M.-G. Ascenzi, The restricted tangent bundle of a rational curve on a quadric in P 3 , Proc. Amer. Math. Soc. 98 (1986) 561–572. [5] E. Ballico, B. Russo, On the stability of the restriction of T P n to projective curves, in: Proc. of Complex Anal. and Geom., in: Pitman Res. Notes Math. Ser., vol. 366, Longman Sci. Tech., Harlow, 1997, pp. 7–18. [6] A. Bernardi, Apolar ideal and normal bundle of rational curves, arXiv:1203.4972 [math.AG], 22 March 2012. [7] A. Bernardi, Normal bundle of rational curves and Waring decomposition, J. Algebra 400 (2014) 123–141, https://doi.org/10.1016/j.jalgebra.2013.11.008. [8] A. Bernardi, A. Gimigliano, M. Idà, On parametrizations of plane rational curves and their syzygies, Math. Nachr. 289 (2016) 537–545, https://doi.org/10.1002/mana.201500264. [9] D.A. Cox, A.A. Iarrobino, Strata of rational space curves, Comput. Aided Geom. Design 32C (2015) 50–68, https://doi.org/10.1016/j.cagd.2014.11.004. [10] D.A. Cox, T.W. Sederburg, F. Chen, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Design 15 (1998) 803–827, https://doi.org/10.1016/S0167-8396(98)00014-4.

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