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SPARSE PARAMETRIZATION OF PLANE CURVES TOBIAS BECK AND JOSEF SCHICHO Abstract. We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry.

Contents 1. Introduction 2. Preliminaries on Toric Varieties 2.1. Embedding the curve into a complete toric surface 2.2. Toric invariant divisors 2.3. Linear systems of toric invariant divisors 3. Sparse Parametrization 3.1. The genus 3.2. The parametrizing linear system 3.3. The algorithm 3.4. An example 4. Another Proof of Correctness 4.1. Some exact sequences 4.2. The sheaf on the normalized curve 4.3. Reduction of the parametrization problem to rational normal curves 4.4. Vanishing of the first cohomology 5. Conclusion References

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Date: June 6, 2005. The authors were supported by the FWF (Austrian Science Fund) in the frame of the research projects SFB 1303 and P15551. 1

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TOBIAS BECK AND JOSEF SCHICHO

1. Introduction Given a bivariate polynomial f ∈ K[x, y] over some field K we will describe a method to find a proper parametrization of the curve defined implicitly by f . That is we will find (X(t), Y (t)) ∈ K(t) s.t. f (X(t), Y (t)) = 0 and (X(t), Y (t)) induces a birational morphism from the curve to the affine line. This is the problem of finding a rational parametrization, a well-studied subject in algebraic geometry. There are already several algorithms, e.g. [10] and [14]. But up to now these methods do not take into account whether the defining equation is sparse or not. We will present an algorithm which exploits the shape of the Newton polygon of the defining polynomial by embedding the curve in a well-chosen complete surface. In this article we do not care for the involved extensions of the coefficient field and therefore assume that K is algebraically closed. This article is organized as follows. In section 2 we recall some basic constructions of toric geometry. In particular we show how to embed a curve into a suitable complete toric surface. In section 3 we show how to compute the genus of the curve in this setting and how to find a linear system of rational functions on the curve that allows to find a parametrization. We state the main theorem which proves the algorithm to be correct. Finally we give a coarse description of the algorithm in pseudo-code and an example. The last section is devoted to a different proof of correctness using sheaf theoretical and cohomological arguments. The reason for giving two different proofs is a historical one. When constructing the algorithm, we were looking for a suitable vector space of rational functions for the parametrization map. We found it by computing the first cohomology of certain sheaves of rational functions. Afterwards it turned out that more elementary arguments (using only the notion of divisors) can also be used to prove correctness of the algorithm. So in one sense the second proof is redundant. We decided to keep it nevertheless in the hope that it provides additional insight. When not seen in another context, the fact that a certain vector space has exactly the right dimension looks like a nice coincidence. 2. Preliminaries on Toric Varieties Let K be an algebraically closed field. We are going to parametrize (if possible) a plane curve F 0 that is given by an absolutely irreducible polynomial f ∈ K[x, y] on the torus T := (K∗ )2 . Actually we will study a curve F which is the Zariski closure of F 0 in a complete surface containing the torus, i.e. F ∩ T = F 0 . We will first show how to realize this surface and then recall some basic definitions and propositions. A good introduction to toric varieties can be found in [2]. 2.1. Embedding the curve into a complete toric surface. Parametrization by rational functions is a “global problem”. In order to apply some theorems of global content we have to embed T in a complete surface. One possible complete surface, which is often used in this context, is the projective plane P2K . We will choose a complete toric surface instead, whose construction is guided by the shape of the Newton polygon of f . In fact also P2K is a complete toric surface and corresponds to the Newton polygon of a dense polynomial f . The Newton polygon Π(f ) ⊂ R2 is defined as the convex hull of all lattice points (r, s) ∈ Supp(f ) (i.e. all (r, s) ∈ Z2 s.t. xr y s appears with a non-zero coefficient in f ). For instance, if f is dense of total degree d then the Newton polygon is equal

SPARSE PARAMETRIZATION OF PLANE CURVES

s

3

v5 = v 6

v7 = v 0

v4 Π(f ) v3

(a2 , b2 ) e2 v1

v2 h2 r

Figure 1. Newton polygon Newton polygon Π(f ) with n = 7 vertices vi . We emphasize edge e2 and show its normal vector (a2 , b2 ) = (−1, 1) and the support half plane h2 . Here we would get c2 = −3. The procedure of lemma 1 inserts an extra normal vector and thus we have the “double” vertex v 5 = v6 .

to the triangle with vertices (0, 0), (d, 0) and (0, d). A sample Newton polygon is illustrated in figure 1. Throughout this article we will always implicitly assume that Π(f ) is non-degenerate; if f is irreducible and Π(f ) is a line segment or a point then the support of f has cardinality at most 2 and the parametrization problem is trivial. For any pair of relatively prime a, b ∈ Z, let c(a, b) ∈ Z be the minimum value of ar + bs, where (r, s) ∈ Π(f ). Then Π(f ) is a finite intersection of say n support half planes hi := {(r, s) ∈ R2 | ai r + bi s ≥ ci } where ci := c(ai , bi ). We assume that the (ai , bi ) are arranged cyclically, i.e. ai−1 bi − ai bi−1 > 0 (setting a0 := an and b0 := bn ). We also give names to the edges and the vertices of intersection ei := {(r, s) ∈ R2 | ai r + bi s = ci } and vi := ei ∩ ei−1 . Note that the set of half planes is not uniquely defined, there may be redundant half planes where the edge meets Π(f ) in one vertex (in this case, some of the vertices vi will coincide). Lemma 1. We can assume that ai−1 bi − ai bi−1 = 1 for 1 < i ≤ n. Proof. The values ai−1 bi − ai bi−1 are invariant under unimodular transformations (i.e. linear transformations by an integral matrix with determinant 1). Assume that ai0 −1 bi0 − ai0 bi0 −1 > 1 for some i0 . By a suitable unimodular transformation we may assume (ai0 , bi0 ) = (0, 1). It follows that ai0 −1 > 1.

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We insert a new index, for simplicity say i0 − 12 , set ai0 − 12 := 1 and determine bi0 − 21 by integer division s.t. 0 ≤ ai0 −1 bi0 − 21 − bi0 −1 < ai0 −1 . It follows ai0 − 21 bi0 − ai0 bi0 − 21 ai0 −1 bi0 − 12 − ai0 − 21 bi0 −1

= 1 · 1 − 0 · bi0 − 21 = ai0 −1 bi0 − 21 − 1 · bi0 −1 = ai0 −1 · 1 − 0 · bi0 −1

= 1 and < ai0 −1 = ai0 −1 bi0 − ai0 bi0 −1 .

By inserting the additional support half plane with normal vector (ai0 − 12 , bi0 − 21 ) and support line through the vertex vi0 , we “substitute” the value ai0 −1 bi0 −ai0 bi0 −1 by the smaller value ai0 −1 bi0 − 12 − ai0 − 21 bi0 −1 and add ai0 − 12 bi0 − ai0 bi0 − 12 = 1 to the list. All other values stay fixed. Repeating this process the statement in the proposition can be achieved. ˜i := A2 (again Now we construct the toric surface. Let 1 ≤ i ≤ n and set U K ˜ ˜ identifying U0 and Un ) with coordinates ui and vi . We denote the coordinate axes ˜i | ui = 0} and Ri := {(ui , vi ) ∈ U ˜i | vi = 0} and define an by Li := {(ui , vi ) ∈ U open embedding of the torus φi : T → A2K : (x, y) 7→ (ui , vi ) = (xbi y −ai , x−bi−1 y ai−1 ).

˜i \ (Li ∪ Ri ) and on Ti the morphism φi has the Its isomorphic image is Ti = U inverse a

b

(ui , vi ) 7→ (x, y) = (ui i−1 viai , ui i−1 vibi ).

For 1 ≤ i ≤ n we define isomorphisms

˜i−1 \ Ri−1 → U ˜i \ Li : (ui−1 , vi−1 ) 7→ (ui , vi ) = (uai−2 bi −ai bi−2 vi−1 , u−1 ) ψi−1,i : U i−1 i−1

with inverses a

−1 ψi,i−1 := ψi−1,i : (ui , vi ) 7→ (ui−1 , vi−1 ) = (vi−1 , ui vi i−2

bi −ai bi−2

).

For two non-neighboring indices i and j, we get isomorphisms ψi,j := φj ◦ φ−1 : i Ti → Tj . The ψi,j then satisfy the gluing conditions ψj,k ◦ ψi,j = ψi,k whenever both sides are defined. Hence they describe an abstract variety V , the toric surface defined by Π(f ). ˜i corresponds to an isomorphic open subset Via the gluing construction, each U Ui ⊂ V which together cover V . For any index i the open subset Ui−1 ∩ Ui cor˜i−1 \ Ri−1 and U ˜i \ Li that are isomorphic by responds to the two open subsets U ˜i−1 and Li ⊂ U ˜i is a curve ψi−1,i . The union of the sets corresponding to Ri−1 ⊂ U 1 in V isomorphic to PK which we call edge curve and denote by Ei . The curves Ei−1 ˜i . For and Ei intersect transversally in a point Vi ∈ Ui , corresponding to (0, 0) ∈ U non-neighboring indices i and j the edge curves Ei and Ej are disjoint. The complement of the union of all edge curves is the torus T , which is also the intersection of all open sets Ui . Now the curve F given by f is defined to be the Zariski closure of F 0 in V . We will see its local equations in the next section. For the rest of the article we fix f and the corresponding curve F ⊂ V , i.e. in particular the data ai , bi , ci derived from its Newton polygon. Remark 2. The proof of lemma 1 corresponds to the resolution procedure for toric surfaces. Being covered by affine planes A2K the constructed toric surface is actually smooth. For the parametrization algorithm this is not strictly necessary (although


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it may be useful). Also the theoretical arguments that follow do not really need a smooth surface. Everything would go through as well using affine toric charts. We use this construction mainly for simplifying notation. 2.2. Toric invariant divisors. Irreducible curves on V are also called prime (Weil) divisors. A general divisor D is defined to be a formal sum (i.e. a linear combination over Z) of prime divisors. The set of divisors forms a free Abelian group Div(V ). The edge curves Ei of the previous section are the special “toric invariant” prime divisors. Consequently a toric invariant divisor is a formal sum of the Ei . One associates to a rational function g ∈ K(x, y) its principal divisor (g). Roughly speaking g has poles and zeros on V along certain subvarieties of codimension one; then (g) is the divisor of zeros minus the divisor of poles (with multiplicities). For a detailed introduction to divisors we refer to [12]. Two divisors are called linearly equivalent iff they differ only by a principal divisor. The divisor class group is defined as the group of divisors modulo this equivalence. The coordinate ring of the torus is K[x, y, x−1 , y −1 ] which is a unique factorization domain. Hence any divisor on the torus is a principal divisor and the class group is trivial. This implies that the class group of the surface V is generated by the toric invariant divisors Ei . We will now show how to find representants in terms of these divisors. Lemma 3. Let g ∈ K[x, y, x−1 , y −1 ]. Then g defines a curve on the torus T . Let G P be its closure in the surface V . Then the divisor G is linearly equivalent to G 0 = ci Ei where c˜i = min(r,s)∈Supp(g) (ai r + bi s), more precisely G = G0 + (g). 1≤i≤n −˜ Proof. On the affine open subset Ui let

−˜ ci−1 −˜ a b vi ci g(ui i−1 viai , ui i−1 vibi ).

gi (ui , vi ) := ui

−˜ c

Then gi ∈ K[ui , vi ] is the local equation of G. Further gi differs from ui i−1 vi−˜ci only by the rational function g which is the same on each affine open set. Together we see that G = G0 + (g). This result holds for any g ∈ K[x, y, x−1 , y −1 ] and of course in particular for g = f , G = F and c˜i = ci from section 2.1. From the proof we get the local equations of the curve F embedded in V (1)

b a −ci−1 −ci vi f (ui i−1 viai , ui i−1 vibi )

fi (ui , vi ) := ui

P and a linearly equivalent divisor F0 := 1≤i≤n −ci Ei . Note that the divisor G0 in the proposition depends only on the support of g, so we define:

DefinitionP4. Given a lattice polygon Π ∈ R2 . We define the associated divisor ci Ei with c˜i = min(r,s)∈Π (ai r + bi s). div(Π) := 1≤i≤n −˜ With this definition of course div(Π(g)) = G0 . In the sequel we will mainly deal with toric invariant divisors.

2.3. Linear systems of toric invariant divisors. A divisor D is called effective (or greater or equal to 0) iff it is a non-negative sum of prime divisors. The linear system of rational functions associated to a divisor D is the vector space of rational functions g ∈ K(x, y) s.t. D + (g) is effective. If D is in particular a toric invariant divisor, then (g) must not have any poles on the torus. Thus we define:

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Definition 5. Given a toric invariant divisor D ∈ Div(V ), we define the linear ˜ system L(D) := {g ∈ K[x, y, x−1 , y −1 ] | (g) + D ≥ 0}. As a corollary to lemma 3 we see that if the divisor is given by a lattice polygon this linear system is non-empty and has a simple description. P Corollary 6. Let Π ∈ R2 be a lattice polygon, let D = div(Π) = 1≤i≤n −˜ c i Ei T 2 r s ˜ and define Π := 1≤i≤n {(r, s) ∈ R | ai r + bi s ≥ c˜i }. Then L(D) = hx y i(r,s)∈Π as a K-vector space.

Here Π is the smallest polygon containing Π and given by an intersection of translates of the half planes hi . 3. Sparse Parametrization

It is well-known that a curve is parametrizable iff it has genus 0. In this section we will first show how to compute the genus in our setting. Afterwards we give a linear system of rational functions that is used to find the parametrizing map. Finally we describe the algorithm and execute it on an example. 3.1. The genus. If the curve F was embedded in the projective plane P2K and had P − P ∈C δP . total degree d , we would have the genus formula g(C) = (d−1)(d−2) 2 The number δP is a measure of singularity, which is defined as the dimension of the quotient of the integral closure of the local ring by the local ring at P (cf. [7, exercise IV.1.8]). For instance, if P is an ordinary singularity of multiplicity µ, i.e. a self-intersection point where µ branches meet transversally, then δP = µ(µ−1) . In 2 particular the sum may be restricted to range over all singular points P ∈ C. If Π ⊂ R2 is a bounded domain we denote by #(P i) := |Π ∩ Z2 | the number of lattice points in Π. We also write Π◦ for the interior of a domain. In the toric situation the genus can be computed as follows: Proposition 7. The genus of F is equal to the number of interior lattice points of Π(f ) minus the sum of the δ-invariants of all points of F : X g(F ) = #(Π(f )◦ ) − δP P ∈F

(The sum actually ranges over the singular points of F only.) Proof. Let F˜ → F be the normalization of the curve. The genus of F can be defined as the arithmetic genus ga (F˜ ) of its normalization. It is known that ga (F˜ ) = P ga (F ) − P ∈F δP (cf. [7, exercise IV.1.8]). The fact that the arithmetic genus ga (F ) equals the number of interior lattice points of Π(f ) is a consequence of the adjunction formula (cf. [4, p. 91]). 3.2. The parametrizing linear system. First we define a family of special divisors on V . Definition 8. Choose 1 ≤ m < n and let ci if 1 ≤ i ≤ m and c0i = ci + 1 else

where the ci originate from the lattice polygon Π(f ). We define Dm := F0 − P P 0 E = i m+1≤i≤n 1≤i≤n −ci Ei ∈ Div(V ) and denote by di := #(ei )−1 the number P of lattice points on the edge i. Further we define the constant d˜m := 1≤i≤m di .


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From now on fix such an m. Note that we deliberately excluded m = n. Now we compute the intersection number F0 · Dm , i.e. the number of intersections of F0 and Dm counting multiplicities. Lemma 9. If d˜m ≥ 2 then F0 · Dm = 2#(Π(f )◦ ) + d˜m − 2. Proof. Let vi0 = (ri0 , si0 ) be the vertex of Π(f ) common to the edges ei0 −1 and ei0 . The intersection number is invariant w.r.t. linear equivalence of divisors. Then F0 · E i 0

= (F0 + (xri0 y si0 )) · Ei0 P P = 1≤i≤n (ai ri0 + bi si0 )Ei · Ei0 1≤i≤n −ci Ei + P = 1≤i≤n (−ci + ai ri0 + bi si0 )Ei · Ei0 1)

= (−ci0 −1 + ai0 −1 ri0 + bi0 −1 si0 )Ei0 −1 · Ei0

2)

+ (−ci0 + ai0 ri0 + bi0 si0 )Ei0 · Ei0 + (−ci0 +1 + ai0 +1 ri0 + bi0 +1 si0 )Ei0 +1 · Ei0

= ai0 +1 (ri0 − ri0 +1 ) + bi0 +1 (si0 − si0 +1 )

= h(ai0 +1 , bi0 +1 ), (ri0 − ri0 +1 , si0 − si0 +1 )i

where 1) holds because Ei and Ei0 are disjoint for non-neighboring indices i and i0 and 2) holds because of the choice of (ri0 , si0 ) and Ei0 , Ei0 +1 intersecting transversally. Finally the vector (ri0 − ri0 +1 , si0 − si0 +1 ) is equal to di0 (−bi0 , ai0 ) (because gcd(ai0 , bi0 ) = 1). Computing the scalar product and applying the identity ai0 bi0 +1 − bi0 ai0 +1 = 1 yields F · Ei0 = di0 . Further we compute P F0 · Dm = F0 · F0 − F0 · m+1≤i≤n Ei P 1) = 2V ol(Γ) − m+1≤i≤n di P P 2) = 2#(Π(f )◦ ) − 2 + 1≤i≤n di − m+1≤i≤n di = 2#(Π(f )◦ ) + d˜m − 2

where 1) follows from the self-intersection formula for toric invariant divisors and 2) from Pick’s formula (cf. [4, pp. 111 and 113]). We can also determine the dimension of the associated linear system. ˜ m )) = #(Π(f )◦ ) + d˜m − 1. Lemma 10. We have dimK (L(D Proof. The divisor Dm is associated to a lattice polygon Π which is obtained “by subtracting certain edges of Π(f )”, compare figure 2. For the number of lattice points one verifies the formula P ◦ ˜ #(Π) = #(Π(f )) − m+1≤i≤n di − 1 = #(Π(f ) ) + dm − 1.

But Π is already given by an intersection of translates of the half planes hi . The lemma now follows from corollary 6. ˜ m ) by adding linear constraints We define a subspace of the linear system L(D derived from conductor ideals. Afterwards we state and prove the main theorem.

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s

v5 = v 6

v7 = v 0

v4 Π

d3 = 1 v3 d2 = 1

v1

d1 = 3

v2 r

Figure 2. Dimension of the linear system We illustrate the point counting argument of lemma 10. Let m = 3. The polygon Π corresponding ˜ m ) contains d1 + d2 + d3 − 1 = 4 lattice points more than the interior of Π(f ). to L(D

˜ its integral closure. The set Definition 11. Let R be a commutative ring and R ˜ ⊆ R} C(R) := {r ∈ R | r R ˜ It is called the conductor (cf. [15, chapter V, §5]). is an ideal of both R and R. ˜ m ). For 1 ≤ i ≤ n let fi and ci be as in equation (1) Definition 12. Let g ∈ L(D and let −c0i−1 −c0i b a vi g(ui i−1 viai , ui i−1 vibi )

gi (ui , vi ) := ui

with c0i as in definition 8. We define the linear system of adjoint polynomials by ˜ m ) | gi + hfi i ∈ C(K[ui , vi ]/hfi i) for 1 ≤ i ≤ n}. Am := {g ∈ L(D There are other equivalent ways to define the adjoint system. For example if F has an ordinary singularity P of multiplicity µ then g (resp. one of the gi ) has to vanish on P with multiplicity at least µ − 1. For more complicated singularities also infinitely close neighboring points have to be taken into account. Theorem 13. Assume d˜m ≥ 2. Let g be a generic polynomial in Am and let G ⊂ V be the Zariski closure of the curve defined by g on the torus. Then G and F have d˜m − 2 free intersections (i.e. intersections not determined by the linear constraints) and dimK (Am ) = d˜m − 1. Proof. For being adjoint to F the curve G has to pass through the singularities of F in a certain way and therefore has to meet additional linear constraints. In fact each singularity P poses δP constraints and gives rise to a local intersection of multiplicity 2δP (cf. [5]).


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The number of free intersections of G and F is therefore equal to P F 0 · Dm − P ∈F 2δP P lem. 9 = 2#(Π(f )◦ ) + d˜m − 2 − 2 P ∈F δP P = 2 #(Π(f )◦ ) − P ∈F δP +d˜m − 2. | {z } =0

It is not clear a priori that the linear conditions are linearly independent, but we can compute a lower bound for the dimension of the system: ˜ m )) − P dimK (Am ) ≥ dimK (L(D P ∈F δP P lem. 10 = #(Π(f )◦ ) + d˜m − 1 − P ∈F δP P = #(Π(f )◦ ) − P ∈F δP +d˜m − 1 | {z } =0

In both cases we finally apply the genus formula of proposition 7. The last inequality is actually an equality. Assume indirectly that dim K (Am ) ≥ ˜ dm . In this case we could choose another set {Pj }1≤j≤d˜m −1 ⊂ F of d˜m − 1 smooth points. Restricting the system Am to a system A˜m by requiring that g ∈ A˜m (resp. the corresponding gi ) also has to vanish on each of the Pj , we get dimK (A˜m ) ≥ 1 and intersection number −1. We would have constructed a curve on V (different from F ) with negative intersection number, a contradiction. For another argument we refer to remark 18. This result is the main ingredient of the sparse parametrization algorithm. Assume d˜m ≥ 3. Choosing d˜m − 3 additional smooth points {Pj }1≤j≤d˜m −3 ⊂ F we restrict the system Am to a system A˜m by requiring that g ∈ A˜m (resp. the corresponding gi ) also has to vanish on each of the Pj . We get dimK (A˜m ) = 2 and the number of free intersections drops to 1. Now let {p, q} ⊂ A˜m be a basis, then F and the zero set of p + tq have (for generic values of t) one intersection in the torus depending on t. The coordinates of this intersection point can be expressed as rational functions in t; this is the parametrization. It is birational by construction, the inverse is given by t = −p/q. 3.3. The algorithm. We give the coarse description of an algorithm (algorithm 1) that exploits theorem 13 to compute a parametrization. The first step is to compute the representation of the objects, e.g. the integers ai , bi , ci defining the morphisms φi and the polynomials fi (see equation (1)). The algorithm contains several loops over finite sets of points. The involved computations always have to use these local representations. Line 2 makes the algorithm more economic because d˜m − 3 is the number of additional smooth points that have to be chosen later. Then (lines 3 to 7) we compute the genus, applying proposition 7, in order to decide rationality and return FAIL if the curve is not parametrizable. Now we compute the parametrizing linear system (lines 8 to 13). We start with ˜ m ). In a real implementation this could mean, we make an indetermined Ansatz L(D P ˜ m ), the sum ranging only over a finite g = (r,s) c(r,s) xr y s for a polynomial in L(D number of indices, compare lemma 10 and figure 2.

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Algorithm 1 P arametrize(f : K[x, y]) : K(t)2 ∪ {FAIL}

: an irreducible polynomial f ∈ K[x, y] : a proper parametrization (X(t), Y (t)) ∈ K(t)2 s.t. f (X(t), Y (t)) = 0 or FAIL if no such parametrization exists S Compute Π(f ) and determine the chart representation F ⊂ V = 1≤i≤n Ui ; Find m and renumber indices s.t. d˜m − 2 is minimally positive; δ := 0; for P ∈ Sing(F ) do δ := δ + δP ; if #(Π(f )◦ ) − δ 6= 0 then return FAIL; ˜ m ); S := L(D for P ∈ Sing(F ) do Add to S the adjoint conditions imposed by P ; Choose a set {Pj }1≤j≤d˜m −3 of smooth points on F ; for 1 ≤ j ≤ d˜m − 3 do Add to S the vanishing condition imposed by Pj ; return F indmap(f, S);

Input Output 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

In order to add to S the adjoint conditions, one could compute the Puiseux expansions of the curve branches at the singular points, substitute those expansions into g (or one of the gi respectively, see definition 12) and extract the linear constraints by enforcing the result to vanish with a certain minimum order. In positive characteristic, when Puiseux expansions are generally not available, the adjoint conditions can be determined using Hamburger-Noether expansions (see [1]). Another method would be to compute (locally) the conductor ideal (see [9]), reduce g (resp. gi ) w.r.t. this ideal and extract the linear constraints by enforcing ideal membership. In order to add to S the vanishing conditions for the smooth points, one simply substitutes the coordinates of a Pj into g (resp. gi ) and equates to zero. In the final step we call a procedure F indmap to actually compute the parametrizing map. It could for example choose a basis {p, q} ⊂ S and then solve the zero-dimensional system f = p + tq = 0 in K(t)[x, y] for (x, y) 6∈ K2 (using Gr¨ obner bases or resultants). Remark 14. As mentioned before it is not strictly necessary to carry out the resolution process of lemma 1. But if one does, one can compute locally using bivariate polynomial representations. In this setting some computer algebra systems (e.g. Singular [6] and Maple) provide functions to compute the delta invariants, certain series expansions of plane curves, etc. They can be used to determine the adjoint conditions. 3.4. An example. Consider the curve F defined by the polynomial f := −27y 21 + 8x2 y 18 + 13x3 y 16 − 8x5 y 13 − 4x4 y 14 + 4x7 y 10

− 20x6 y 11 − 8x8 y 8 + 8x10 y 5 + 4x9 y 6 + 8x11 y 3 + 4x13 ∈ Q[x, y]


11

s 20

15

10

Π(f )

5

5

10

15

20 r

Figure 3. Newton polygon of f The Newton polygon Π(f ) of the example is very “slim”. Its only interior points are (3, 16), (5, 13), (6, 11) and (8, 8).

on the torus. Its Newton polygon Π(f ) has 6 vertices. We represent it as the intersection of n = 8 half planes hi which are governed by the following data: v1 = (13, 0),

(a1 , b1 ) = (−5, −3),

d1 = 2

c2 = −106,

d2 = 1

c4 = 21,

d4 = 0

(a5 , b5 ) = (7, 4),

c5 = 84,

d5 = 1

v6 = (4, 14),

(a6 , b6 ) = (5, 3),

c6 = 62,

d6 = 0

v7 = (4, 14),

(a7 , b7 ) = (8, 5),

c7 = 102,

d7 = 1

v8 = (9, 6),

(a8 , b8 ) = (3, 2),

c8 = 39,

d8 = 2

v2 = (7, 10),

(a2 , b2 ) = (−8, −5),

c1 = −65,

(a4 , b4 ) = (2, 1),

c3 = −42,

d3 = 1

v4 = (0, 21),

(a3 , b3 ) = (−3, −2),

v5 = (0, 21),

v3 = (2, 18),

The half planes h4 and h6 have been introduced in order to fulfill the condition ai−1 bi − ai bi−1 = 1, although it will turn out that they do not really play a role here. By doing so, we added the vertices v4 = v5 , v6 = v7 and the edges e4 , e6 of length d4 = d6 = 0. From the Newton polygon (see figure 3) we see that #(Π(f )◦ ) = 4. For this example the half planes have already been arranged s.t. by setting m := 2 we get the optimal value d˜m = 3. Consequently we will not have to choose any smooth points to get additional linear constraints later on. S The curve F is (by our construction) embedded in a toric surface V = 1≤i≤8 Ui . We compute the representation of F in local coordinates using equation (1): f1

= −27v12 u31 − 4v13 u1 + 13v12 u21 + 8v1 u31 − 20v12 u1

− 8v1 u21 + 4v12 − 8v1 u1 + 4u21 + 8v1 + 8u1 + 4

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f2 f3 f4 f5 f6 f7 f8

= −4v24 u32 + 4v24 u22 − 20v23 u22 + 8v23 u2 + 13v22 u22

− 8v22 u2 − 27v2 u22 + 4v22 − 8v2 u2 + 8v2 + 8u2 + 4

= 4v33 u43 + 8v33 u33 − 4v32 u43 + 4v33 u23 − 20v32 u33

− 8v32 u23 + 8v32 u3 + 13v3 u23 − 8v3 u3 + 4v3 − 27u3 + 8

= 4v45 u34 + 8v44 u34 + 8v44 u24 + 4v43 u34 − 8v43 u24

+ 4v43 u4 − 20v42 u24 − 8v42 u4 − 4v4 u24 + 13v4 u4 + 8v4 − 27

= 4v57 u55 + 8v56 u45 + 8v55 u45 + 4v55 u35 − 8v54 u35

+ 4v53 u35 − 8v53 u25 − 20v52 u25 + 8v52 u5 + 13v5 u5 − 4u5 − 27

= 4v63 u76 + 8v63 u66 + 4v63 u56 + 8v62 u56 − 8v62 u46

− 8v62 u36 + 8v62 u26 + 4v6 u36 − 20v6 u26 + 13v6 u6 − 27v6 − 4

= 4v74 u37 + 8v74 u27 + 8v73 u37 − 8v73 u27 + 4v72 u37

− 27v73 u7 − 8v72 u27 + 13v72 u7 + 8v7 u2 − 20v7 u7 + 4u7 − 4

= 8v83 u48 − 27v83 u38 + 4v82 u48 − 8v82 u38 + 13v82 u28

+ 8v8 u38 − 8v8 u28 − 20v8 u8 + 4u28 − 4v8 + 8u8 + 4

It turns out that F has a singular point P on the toric invariant divisor E1 . It shows up in the open subsets U1 and U2 and has coordinates (u1 , v1 ) = (−1, 0), (u2 , v2 ) = (0, −1) respectively. Another singular point Q ∈ E8 is lying in U1 and U8 with coordinates (u1 , v1 ) = (0, −1) and (u8 , v8 ) = (−1, 0). The curves F ∩ Ui for i ∈ {3, 4, 5, 6, 7} are smooth. Hence all the information on the singularities can be gathered in U1 . For this purpose we compute the Puiseux expansions at the points P and Q: σP (α) σQ (α)

= − 41 (u1 + 1) + α(u1 + 1)2 + ( 435 608 α − = −1 − 25 u1 + βu21 + ( 21 4 β+

51 2432 )(u1

+ 1)3 . . .

3 195 16 )u1 . . .

Here α denotes a root of 1024α2 +516α+63 and β denotes a root of 16β 2 +24β −45. Taking conjugates into account we have two curve branches through each of the singular points. From these expansions one can compute amongst others the δ-invariants δP = δQ = 2 (for details we refer to [13]). We compute the genus g(F ) = #(Π(f )◦ ) − δP − δQ = 4 − 2 − 2 = 0, i.e. the curve F is indeed parametrizable. ˜ m ). The support Now we make an indetermined Ansatz for a polynomial in L(D of such a polynomial has to lie within Π(f ) but not on the edges ei for i ∈ {3, . . . , 8}. g := c1 x3 y 16 + c2 x5 y 13 + c3 x7 y 10 + c4 x6 y 11 + c5 x8 y 8 + c6 x10 y 5

For obvious reasons, we only have to compute the local representation in U 1 (see definition 12): g1 = c1 v12 u1 + c4 v12 + c2 v1 u1 + c5 v1 + c3 u1 + c6 In order to be adjoint g1 (σP ) has to vanish with order at least 2 around u1 = −1 and g1 (σQ ) has to vanish with order at least 2 around u1 = 0 (again see [13]). Executing the substitutions and equating lowest terms to 0 one gets the linear constraints 14 c2 + c3 − 14 c5 = 0, −c3 − c6 = 0 (from P ) and c4 − c5 + c6 = 0, c1 − c2 + c3 + 5c4 − 52 c5 = 0 (from Q). We solve this system w.r.t. parameters c3 , c4 and substitute the result into g to get a polynomial g˜ ∈ Am (for any concrete


13

value of c3 and c4 ). g˜ = (− 32 x3 y 16 − 3x5 y 13 + x7 y 10 + x8 y 8 + x10 y 5 )c3

+ (− 32 x3 y 16 + x5 y 13 + x6 y 11 + x8 y 8 )c4

As a final step we solve the system {f = 0, g˜ = 0, c3 = 1, c4 = t} for x and y in Q(t). It has two distinct solutions. One is (x, y) = (0, 0) which corresponds to the two singular points P and Q, the other one yields the parametrization: −256(2t2 + 4t − 1)3 (t + 1)7 t8 (−1 + 8t)3 (2t2 + 7t − 1)5 −32t5 (2t2 + 4t − 1)2 (t + 1)4 Y (t) = (−1 + 8t)2 (2t2 + 7t − 1)3

X(t) =

For illustrative purposes assume we had chosen m := 3 non-optimal (or we were in a situation where a choice s.t. d˜m = 3 is not possible). We would get the following ˜ m ): indetermined Ansatz for a polynomial in L(D g := c0 x2 y 18 + c1 x3 y 16 + c2 x5 y 13 + c3 x7 y 10 + c4 x6 y 11 + c5 x8 y 8 + c6 x10 y 5 We compute the local representations in U1 and U5 . g1 = c1 v12 u1 + c0 v1 u21 + c4 v12 + c2 v1 u1 + c5 v1 + c3 u1 + c6 g5 = c6 v55 u35 + c3 v54 u25 + c5 v53 u25 + c2 v52 u5 + c4 v5 u5 + c0 v5 + c1 Proceeding as before, i.e. substituting the Puiseux expansions at the singular points P and Q in U1 into g1 we get the linear constraints − 41 c0 + 41 c2 + c3 − 14 c5 = 0, −c3 + c6 = 0, c4 − c5 + c6 = 0 and c1 − c2 + c3 + 5c4 − 52 c5 = 0. Now we have to choose an additional smooth point on F , e.g. (u5 , v5 ) = (− 27 4 , 0) in U5 . Plugging these coordinates into g5 and equating the result to zero we get c1 = 0. We again solve the system and substitute into g. g˜ = (−4x5 y 13 − x6 y 11 + x7 y 10 + x10 y 5 )c6

+ (x2 y 18 + 53 x5 y 13 + 23 x6 y 11 + 23 x8 y 8 )c0

Now in the same way as above we arrive at the following parametrization: (135 − 108t + 20t2 )3 (14t − 27)7 (−3 + 2t)8 429981696(16t2 − 18t − 27)5 (t − 2)8 t3 −(135 − 108t + 20t2 )2 (14t − 27)4 (−3 + 2t)5 Y (t) = 248832t2(16t2 − 18t − 27)3 (t − 2)5

X(t) =

Remark 15. A conventional algorithm based on an embedding of the curve in the projective plane P2Q has to work very hard on that example. The corresponding complete curve would have again two singular points, but now the δ-invariants are 119 and 71. These complicated singularities show up only because the structure of the Newton polygon is not taken into account. Proceeding in this setting like we did, the involved linear systems are found as subspaces of a vector space of dimension greater than 200. The excellent Maple implementation of a parametrization algorithm produced around 40 DIN A4 pages of output. (Of course we admit that the chosen example is especially well-fit for our method.)

14


4. Another Proof of Correctness This section is devoted to another proof of correctness using sheaf theoretical and cohomological arguments. Also theorem 13 could be deduced from what follows. From now on let F˜ be the normalization of F . We are in the following situation π ι F˜ F ,→ V

and assume that F is parametrizable, i.e. g(F˜ ) = g(F ) = 0. 4.1. Some exact sequences. The curve F is a closed subscheme of V . Let I(F ) denote its ideal sheaf. We have an exact sequence of sheaves on V : ι#

0 → I(F ) → OV → ι∗ (OF ) → 0

Now we define the conductor ideal sheaf on F , F˜ and V . Let CF denote the sheaf defined by U 7→ C(OF (U )) (see definition 11) and CF˜ := π ∗ (CF ). Observe that π∗ (CF˜ ) ∼ = (CF ) because the conductor is an ideal sheaf on both F and F˜ . Since CF is a subsheaf of OF trivially ι∗ (CF ) is a subsheaf of ι∗ (OF ). Now we define the conductor sheaf CV on the surface by CV (U ) := (ι# )−1 (ι∗ (CF )(U )) for all open U ⊆ V . Clearly CV is a subsheaf of OV containing ker(ι# ) and the restriction of ι# is still surjective. Thus we have an exact sequence 0 → I(F ) → CV → ι∗ (CF ) → 0.

The invertible sheaf L(D) associated to a Weil divisor D on the smooth variety V is a subsheaf of the sheaf of rational functions KV defined locally by L(D)(U ) = {g ∈ KV (U ) | (g) + D |U ≥ 0}

˜ for all open U ⊆ V . This is a sheafified version of the definition 5. In fact L(D) = Γ(V, L(D)). Tensoring with invertible sheaves is exact so we get the exact sequence 0 → I(F ) ⊗ L(Dm ) → CV ⊗ L(Dm ) → ι∗ (CF ) ⊗ L(Dm ) → 0.

Now we define the following sheaf on F˜ :

J := CF˜ ⊗ (ι ◦ π)∗ (L(Dm )) Applying the projection formula (cf. [7, exercise II.5.1]) we see that (ι ◦ π)∗ (J )

= (ι ◦ π)∗ (CF˜ ⊗ (ι ◦ π)∗ (L(Dm ))) ∼ = (ι ◦ π)∗ (CF˜ ) ⊗ L(Dm ) ∼ = ι∗ (CF ) ⊗ L(Dm ).

∼ L(−F0 ) we have I(F ) ⊗ L(Dm ) ∼ g g Since I(F ) = P L(−F ) = = L(D m ) with Dm := Dm − F0 = − m+1≤j≤n Ej . Putting things together, we get

(2)

g 0 → L(D m ) → CV ⊗ L(Dm ) → (ι ◦ π)∗ (J ) → 0.

Finally the global sections functor is left-exact which yields (3)

g ˜ 0 → Γ(V, L(D m )) → Γ(V, CV ⊗ L(Dm )) → Γ(V, (ι ◦ π)∗ (J )) = Γ(F , J ).

g But D m is the inverse of an effective divisor and consequently has no global sections, g i.e. Γ(V, L(D m )) = 0. In other words: (4) Γ(V, CV ⊗ L(Dm )) ,→ Γ(F˜ , J )


15

The global sections Γ(V, CV ⊗ L(Dm )) are very suitable for computation. Indeed if we write CV ⊗ L(Dm ) as a sheaf of rational functions, we see that its global sections correspond exactly to the system Am of definition 12. In fact the last map of sequence (3) is also surjective and thus (4) is an isomorphism. We postpone the cohomological proof of this statement to the last section. Instead we proceed now with a brief study of the sheaf J and interpret the isomorphism in the context of the parametrization problem. 4.2. The sheaf on the normalized curve. First we reinterpret lemma 9 in the context of sheaves. In general for any divisor D ∈ Div(V ) it is true that deg((ι ◦ π)∗ (L(D))) = F · D. Then F · Dm = F0 · Dm implies the following corollary. Corollary 16. deg((ι ◦ π)∗ (L(Dm ))) = 2#(Π(f )◦ ) + d˜m − 2. Proposition 17. If

P

1≤j≤m

dj ≥ 2 then deg(J ) = d˜m − 2.

Proof. We compute the degree of J using deg(CF˜ ) = −2 16 and applying the genus formula of proposition 7: deg(J ) = = = = =

P

P ∈F

δP (cf. [5]), corollary

deg(CF˜ ⊗ (ι ◦ π)∗ (L(Dm ))) deg(CF˜ ) + deg((ι ◦ π)∗ (L(Dm ))) P −2 P ∈F δP + 2#(Γ(f )◦) + d˜m − 2 P 2 #(Γ(f )◦ ) − P ∈F δP + d˜m − 2 d˜m − 2.

Assume d˜m ≥ 3 and let d := d˜m − 2. Since F is assumed parametrizable, its normalization F˜ is isomorphic to P1K . Assume we have homogeneous coordinates u, v on P1K . Let P ∈ Div(P1K ) be the prime divisor corresponding to u = 0. Any invertible sheaf of degree d on P1K is isomorphic to L(dP ). This sheaf is generated by its global sections Γ(P1K , L(dP )) = hv d /ud , v d−1 u/ud, . . . , ud /ud i. They constitute a closed immersion ψ : P1K → PdK : [u : v] 7→ [v d : v d−1 u : · · · : ud ]. In other words, a basis of the global section space Γ(F˜ , J ) defines an isomorphism between F˜ and the rational normal curve in PdK . Remark 18. From this one could also get a slightly different proof of theorem 13 because deg(J ) corresponds to the number of free intersections. The result on the dimension follows from the above arguments because dimK (Am ) = dimK (Γ(V, CV ⊗ L(Dm ))) = dim(Γ(F˜ , J )) = deg(J ) + 1. 4.3. Reduction of the parametrization problem to rational normal curves. Write CV ⊗ L(Dm ) as a sheaf of rational functions and identify J with L(dP ) on P1K as above. The functions in Γ(V, CV ⊗ L(Dm )) do not have a pole along F . The reader may check that isomorphism (4) is in fact given by the pullback (ι ◦ π)∗ of rational functions. Now let {s0 , . . . , sd } ⊂ Γ(V, CV ⊗ L(Dm ))) be a basis s.t. (ι ◦ π)∗ (si ) = v d−i ui /ud and define a rational map by φ : T → PdK : (x, y) 7→ [s0 : s1 : · · · : sd ]

16


on the torus. We find that it maps F˜ (and hence also F ) birationally to the rational normal curve in PdK : ι◦πF˜ ∼ V = P1K ψ

-

φ ? PdK

In the algorithm we finally choose a set of d˜m − 3 = d − 1 smooth points and restrict the linear system using vanishing conditions imposed by these points. In our current setting this corresponds to choosing points on the rational normal curve and projecting until we reach the projective line P1K ; the natural way to parametrize a rational normal curve. 4.4. Vanishing of the first cohomology. It remains to show that the last map ˇ of sequence (3) is surjective. We will use Cech cohomology w.r.t. the natural affine cover U := {Ui }1≤i≤n to derive the desired result. From the short exact sequence (2) we get a long exact sequence 0 → →

g Γ(V, L(D → Γ(V, CV ⊗ L(Dm )) → Γ(V, (ι ◦ π)∗ (J )) m )) 1 ˇ g H (U, L(Dm )) → ...

ˇ 1 (U, L(D g So we have to show that H m )) = 0. ˇ To define Cech cohomology we use the following set of half planes: 0 for 1 ≤ i ≤ m and 2 ¯ hi := {(r, s) ∈ R | ai r + bi s ≥ ∆i } where ∆i = 1 else

Using the coordinate transformations of section 2 and these half planes one describes g the needed sections of L(D m ) as K-vector spaces: g Γ(Ui , L(D m ))

g Γ(Ui−1 ∩ Ui , L(D m )) g Γ(Ui ∩ Uj , L(D m ))

g Γ(Ui ∩ Uj ∩ Uk , L(D m ))

= hxr y s i(r,s)∈h¯ i−1 ∩h¯ i for 1 ≤ i ≤ n,

= hxr y s i(r,s)∈h¯ i for 2 ≤ i ≤ n,

= hxr y s i(r,s)∈Z2 for 1 ≤ i < j ≤ n and j − i ≥ 2, = hxr y s i(r,s)∈Z2 for 1 ≤ i < j < k ≤ n

ˇ The first objects in the Cech complex are given by Y g g Γ(Ui , L(D C0 (U, L(D m )), m )) = 1≤i≤n

g C (U, L(D m )) 1

g C (U, L(D m )) 2

and the first maps by

=

Y

1≤i