Plane polar Cremona maps of arbitrarily large degree in positive characteristic

A result of I.V.Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of degree at most three. In this note we describe homaloidal polynomials in three variables of arbitrarily large degree in positive characteristic. Using combinatorial arguments, we also classify line arrangements whose polar map is homaloidal in positive characteristic.


Introduction
Given a homogeneous polynomial f ∈ k[x 0 , . . . , x m ] over a field k, the polar map Φ f : P m k P m k of f is the rational map defined by the linear system ∂f ∂x0 , . . . , ∂f ∂xm . The polynomial f is called homaloidal if ∂f ∂x0 , . . . , ∂f ∂xm has no fixed component and Φ f is birational.
It was established by I.V. Dolgachev [Dol00,Theorem 4] that if k = C, the homaloidal polynomials in three variables are either of degree 2, defining a smooth conic in the projective plane, or of degree 3, defining either a union of three lines in general position or a union of a smooth conic with one of its tangents. These polynomials remain homaloidal when the base field k has characteristic greater than 2. This leads to the following question which is a generalisation of [DHS12, Question 3.7].
Problem 1. Over a field k of positive characteristic, are there other homaloidal polynomials than the ones in Dolgachev's classification?
In [BC18,Proposition 4.6], a first example of a homaloidal polynomial of degree 5 over a field of characteristic 3 was produced, answering both Problem 1 and [DHS12, Question 3.7]. Very recently, the following example of a homaloidal curve of degree 5 in characteristic 3 was also described.

Problem 3. Does there exist homaloidal polynomials in three variables of arbitrary large degree over fields of arbitrarily large characteristic?
In this note, we answer this question, see Theorem 2.4 for our expanded result. Following the designation in [Hir83] we say that a union of n distinct lines through a given point z 0 with another line not passing through z 0 is a near-pencil arrangement of n + 1 lines. In addition, given a reduced projective curve F = V(f ) which is the zero locus V(f ) in P 2 k of a homogeneous polynomial f , we say that F is homaloidal if f is homaloidal.
Theorem A. Let k be a field of characteristic p and let n ∈ N >0 be a multiple of p. Then the near-pencil arrangement of n + 1 lines is homaloidal.
This result provides an answer to Problem 3. For instance, let n ∈ N >0 be such that n ≡ 0 mod 5 and let k be a field of characteristic 5. Then the polynomial f n in Theorem A has degree n + 1 and is homaloidal. Moreover, given a prime number p, a field k of characteristic p and a positive integer m, Theorem A gives a homaloidal polynomial of degree mp + 1, so homaloidal polynomials exist in arbitrarily large degree and in any prime characteristic.
Remark that the base ideal of the polar map Φ f , i.e. the ideal generated by the partial derivatives of a polynomial f defines the singular locus of the curve V(f ) defined by f in P 2 k . In this direction, the proof given by I.V.Dolgachev about the classification of homaloidal complex polynomials relies on the Jung-Milnor's formula over C relating several invariants of singularities [Dol00, Lemma 3]. In contrast, our proof of Theorem A relies on the study of the torsion of the symmetric algebra of the base ideal of Φ f , an approach that fits in line with previous works such as [RS01], [DHS12], and [BC18]. We emphasize that most of the polynomials we consider in this note define free curves (a curve being free if, by definition, the base ideal of the polar map is determinantal [Dim17a, Def 2.1]). We specially focus on this case since, when the base ideal has a linear syzygy, free curves are the curves whose singular schemes have maximal length [Dim17a, Cor 1.2].
Contents of the paper. In the first section, we recall the relations between the symmetric algebra of the base ideal of a rational map Φ and the graph of Φ. Birationality of a map can be checked via its graph which explains the strategy of detecting a birational map via the symmetric algebra of its base ideal.
The second section constitutes the heart of our work. As a central idea, one can study the reduction modulo p of the presentation matrix of the base ideal in order to predict a drop of the topological degree of the polar map, see Subsection 2.1. The next step is then to evaluate this drop. We carry on this evaluation by describing the generic fibre of the naive graph and thus describing the generic fibre of the graph itself. This implies in particular that the polynomials f n are homaloidal (Lemma 2.3). We end this section by providing another example of a polynomial of degree 5 which is homaloidal in characteristic 3. Its zero locus in P 2 k is the union of the unicuspidal ramphoïd quartic and the tangent cone at its cusp (Example 2.6).
In the third section, we focus on line arrangements (that is plane curves which are union of lines) and we show that, over an algebraically closed field of characteristic p > 0, the only homaloidal line arrangements are the ones defining unions of three general lines or near-pencil of lp + 1 lines for any l 1, see Proposition 3.1. This classification follows from the description of the singularities defined by line arrangements.
The explicit computations given in this paper were made using basic functions of the software systems Polymake and Macaulay2 with the Cremona package [Sta17] associated. The corresponding codes are available on request.
Acknowledgements. I warmly thank Adrien Dubouloz and Daniele Faenzi for useful remarks and suggestions about early versions of this note. I also thank deeply the anonymous referee for having raised the question of the classification of homaloidal line arrangements and for having pointed out a mistake about the computation of Milnor number of singularities in a previous version of this paper.

Graph and naive graph
In this note, all the fields are assumed to be algebraically closed and denoted by the same letter k.
1.1. Multidegree of a subscheme of P 2 × P 2 , projective degrees. For this presentation, we follow [Dol11, 7.1]. Given l ∈ {0, 1, 2}, we denote by H l a general codimension l linear subspace of P 2 k . For a subscheme X ⊂ P 2 k × P 2 k of codimension 2, the multidegree d 0 (X), d 1 (X), d 2 (X) is defined by : k × P 2 k → P 2 k and p 2 : P 2 k × P 2 k → P 2 k are the first and second projection respectively.
Consider now a rational map Φ = (φ 0 : φ 1 : φ 2 ) : P 2 k P 2 k with base ideal where φ 0 , φ 1 , φ 2 are homogeneous polynomials of the same degree that do not share any common factor. The graph Γ of Φ is by definition the closure of {(x, Φ(x)), x ∈ P 2 k \V(I)} ⊂ P 2 k × P 2 k in the Zariski topology. It is an irreducible variety of codimension 2. We define the projective degrees Since Φ is birational if and only if d 0 (Φ) = 1, this last quantity has a special importance and is called the topological degree of Φ.
be the Rees algebra of I. By [Dol00, 7.1.3] the blow-up Proj R(I) of P 2 k with respect to I is the graph Γ of Φ. Moreover, R(I) is the epimorphic image of the symmetric algebra S(I) of I via the epimorphisms I ⊗i ։ I i . Hence, the ideal of the graph Γ ⊂ P 2 k × P 2 k of Φ contains the ideal of P(I) = Proj S(I) ⊂ P 2 k × P 2 k generated by the entries of the matrix y 0 y 1 y 2 M S where S = R[y 0 , y 1 , y 2 ] stands for the coordinate ring of P 2 k × P 2 k and M S stands for a presentation matrix of I ⊂ R tensored with S [BC18, Subsection 1.1] (note that we use the same notation M and M S from now on). The ideal I is said to be of linear type if Γ = P(I) [Vas05, 1.1 Ideals of linear type].
Definition 1.1. The naive graph of Φ is the projectivization P(I) = Proj S(I) of the symmetric algebra of I.
x 2 ] of height at least 2 and where φ 0 , φ 1 , φ 2 have the same degree d. Note that, in this setting, the condition height(I) ≥ 2 is equivalent to the fact that (φ 0 , φ 1 , φ 2 ) do not share any common factor, a property which we always assume in the following. Assume moreover that Φ is determinantal, i.e. that the polynomials φ 0 , φ 1 , φ 2 are the 2-minors of a given 3 × 2-matrix such that all its entries in the first column are homogeneous of degree a and all its entries in the second column are homogeneous of degree b (hence a + b = d). The Hilbert-Burch theorem [Eis95,Theorem 20.15] implies that I has a free resolution of the form: where M is a 3 × 2-matrix with entries in R. Hence M is a presentation matrix of I and P(I) is the intersection of two divisors of P 2 k × P 2 k of bidegree (a, 1) and (b, 1) respectively. Considering the case where I is of linear type, since Γ has codimension 2, P(I) = Γ is a complete intersection and the projective degrees of Φ are given by Bézout's theorem: k be a determinantal rational map with base ideal I = (φ 0 , φ 1 , φ 2 ) of height 2. The naive projective degrees d 0 (Φ), d 1 (Φ), d 2 (Φ) of Φ are defined by the multidegree d 0 (P(I)), d 1 (P(I)), d 2 (P(I)) of its naive graph P(I).
Example 1.4. If I is not necessarily of linear type in Example 1.2, we still have that the naive projective degree of Φ are (ab, a+b, 1) because P(I) is still a complete intersection. However, if there is an extra part T = P(I)\Γ in P(I) with support as in (1.3), we have then depends moreover on the scheme structure of T and is the object of Subsection 2.2.

Contribution of the torsion
In this section, following the situation described in Example 1.4, we illustrate first on an example how to estimate the drop of the topological degree in positive characteristic compared to characteristic 0. We analyse then in greater generality how this modification impacts the computation of the topological degree of Φ.

Reduction of the presentation matrix modulo p.
In what follows, for a homogeneous ideal I of R = k[x 0 , x 1 , x 2 ] and an integer t, we denote by I t the homogeneous piece of I of degree t. Let n ∈ N >1 and let F n be the union of n distinct lines through a point z 0 ∈ P 2 k with any other line not passing through z 0 . We can reduce to the situation where V(x 2 ) is the latter line and two lines among the n th firsts are V(x 0 ) and V(x 1 ) so that z 0 = (0 : 0 : 1). We can consequently assume without loss of generality that an equation of F n reads f n = x 0 x 1 l 2 · · · l n−1 x 2 where, for all i ∈ {2, . . . , n − 1}, l i belongs to (x 0 , x 1 ) 1 (set f n = x 0 x 1 x 2 if n = 2). The ideal I of partial derivatives of f n is then equals to x 0 l 2 · · · l n−1 x 2 + x 0 x 1 ∂ ∂x 1 (l 2 · · · l n−1 )x 2 , x 0 x 1 l 2 · · · l n−1 ).
The ideal I has depth 2 for otherwise the first two generators of I would be divisible by either x 0 , x 1 or l i for i ∈ {2, . . . , n − 1} which is excluded by the assumption that all the lines in F n are distinct. A direct computation shows that where, given j ∈ {1, 2, 3}, M j stands for (−1) j times the minor obtained from M by leaving out the j th row (in order to check these equalities, remark that, for any i ∈ {2, . . . , n − 1}, l i = x 0 ∂li x0 + x 1 ∂li x1 ). Hence I is a determinantal ideal given by the 2-minors of M . Since it has the expected depth, the Hilbert-Burch theorem asserts that a free resolution of I reads: M Moreover, since it does not have constant entries, M is a minimal presentation matrix of I.
Proposition 2.2. Let n ∈ N >1 and k be an algebraically closed field such that p = char(k) does not divide n. Then I is of linear type and the polar map Φ fn of f n has multidegree (n − 1, n, 1).
Proof. Since M has one column of linear entries and one column of entries of degree n, the naive multidegree is (n − 1, n, 1), see Example 1.2. Moreover Fitt 2 (I) = (x 0 , x 1 , x 2 ) is not supported on any point of P 2 k so, by (1.3), I is of linear type. Hence the graph Γ and the naive graph P(I) coincide, so the projective degrees and the naive projective degrees of Φ fn coincide.
We consider now the case where char(k) divides n in a more general situation.
Moreover since all the entries of M are in the ideal (x 0 , x 1 ), the radical Fitt 2 (I) of the ideal Fitt 2 (I) of entries of M is contained in (x 0 , x 1 ). Actually, since I has height 2 and the polynomials in the first column of M are linear in x 0 and x 1 one has Fitt 2 (I) = (x 0 , x 1 ). Hence, as previously stated in (1.3), the naive graph P(I) = V(I P(I) ) ⊂ P 2 x × P 2 y is the union of a torsion part T supported on V(x 0 , x 1 ) = {(0 : 0 : 1)} × P 2 k , and of the graph Γ = P(I)\V(x 0 , x 1 ) of the map Φ whose base ideal is by the 2-minors ideal I of M . The next result is a consequence of [BCRD20, Theorem 5.14] but we will give a self-contained proof. Proof. We analyse separately each element of the multidegree d 0 (Γ), d 1 (Γ), d 2 (Γ) .
We have the following extension of Theorem A.
(1) Let n ∈ N >1 and assume that p = char k divides n, then the near-pencil arrangements of n + 1 lines is homaloidal.
(1) By Lemma 2.1, a presentation matrix of the ideal I of partial derivatives of f n verifies the conditions of Lemma 2.3. Hence Φ fn is birational and since the associated linear system has no fixed component, the polynomial f n is homaloidal. x 2 ) has presentation matrix Indeed, I has height 2 for otherwise x 0 or x 1 would divide x n 1 + (n − 1)x n−2 0 x 1 x 2 and nx 0 x n−1 1 + x n−1 0 x 2 which is not the case. Moreover where given j ∈ {1, 2, 3}, M j is equal to (−1) j times the minor obtained from M by leaving out the j th row. Hence I is a determinantal ideal and, by application of Hilbert-Burch theorem, M is a minimal presentation matrix of I. Now, if p divides n(n − 1) − 1, the matrix M verifies the conditions of Lemma 2.3. So, in this case, Φ gn is birational and g n is homaloidal.
Remark 2.5. The method of reduction modulo p we just described also applies to Example 2 and to the quintic Q 5 = V x 0 (x 2 1 + x 0 x 2 )(x 2 1 + x 0 x 2 + x 2 0 ) described in [BC18].
2.3. Limits and perspectives. The fact that the presentation matrix of the jacobian ideal reduces well modulo p does not always occur, as illustrated by the following example.
. Its zero locus in P 2 k is the union of the unicuspidal ramphoïd quartic with the tangent cone at its cusp, see [Moe08]. Over a field k of characteristic 0, a computation with Macaulay2 shows that a presentation matrix of the ideal of partial derivatives of h reads: We can a priori not expect to apply Lemma 2.3 after reduction modulo p. However, after reducing modulo 3, a presentation matrix of the reduction of I modulo 3 reads   0 This implies that the polar map of h is birational by Lemma 2.3 (here, remark that the torsion is supported on V(x 1 , x 2 ) and that the maximal power of x 0 is 1 is the second column). By application of Hilbert-Burch theorem, we also have that the induced linear system does not have fix components so h is actually homaloidal.
Remark 2.7. As pointed out by Example 2.6 and Item (2) of Theorem 2.4, the classification of homaloidal plane curves in any characteristic seems to be a challenging problem, especially by only looking to the reduction modulo p of the syzygies of the jacobian ideal. One can however restrict first to the classification of homaloidal line arrangements and this is the object of next section.

Classification of homaloidal line arrangements in positive characteristic
As a guideline for the section, let us state first our result about the classification of homaloidal line arrangements.
Proposition 3.1. Given an algebraically closed field k of characteristic p > 0, the only homaloidal line arrangements are: (i) the union of three general lines, (ii) the near-pencils of n + 1 lines where p divides n.
Our proof of Proposition 3.1 mainly relies on the observation that, as far as the topological degree of the polar map of an arrangement is concerned, the only quantity to consider is the numbers of lines defining the singularities of the arrangement. More precisely, a singularity z of a line arrangement A = V(f ) being the intersection of r 2 lines of A, the numerical contribution of z in the computation of d 0 (Φ f ) only depends on whether the characteristic p divides r or not, see Lemma 3.2 for the precise result. Given this fact, the proof of Proposition 3.1 aims to characterize combinatorially near-pencils of p + 1 lines among all arrangements of p + 1 lines and this combinatorial characterization follows from [dBE48, Th.1].
In the following, given an integer d 4, we let f = l 1 · · · l d be the product of d homogeneous linear polynomials l 1 , . . . , l d ∈ k[x 0 , x 1 , x 2 ] and A = V(f ) be the line arrangement defined by f . Moreover, using the designation in [Hir83], a point z in the singular locus of A which is the intersection point of r lines is called a r-fold point.
The field k being algebraically closed, the topological degree d 0 (Φ f ) of Φ f is the degree of the fiber of a generic point of P 2 , that is: where I = ( ∂f ∂x0 , ∂f ∂x1 , ∂f ∂x2 ) is the jacobian ideal of f , I g = (a ∂f ∂x0 +b ∂f ∂x1 +c ∂f ∂x2 , α ∂f ∂x0 + β ∂f ∂x1 +γ ∂f ∂x2 ) is the ideal defined by two generic linear combinations of ∂f ∂x0 , ∂f ∂x1 , ∂f ∂x2 and I g : I ∞ stands for the saturation ideal of I g by I, see [Dol11, 7.1.3] for this computation of the topological degree. Since V(I g : I ∞ ) is set-theoretically equal to V(I g )\V(I), one has thus: where m z is the multiplicity of z in the scheme V(I g ) (note that this latter expression of d 0 (Φ f ) is true for any reduced plane curve V(f ) and not only for line arrangements). Over the field of complex numbers C, by [Dim17b, 4.2], given an r-fold point z ∈ V(I) one has m z = µ f,z = (r − 1) 2 where µ f,z stands for the local Milnor number of A at z, see [Dim17b, Definition 2.17] for the definition of Milnor numbers. Over a field of positive characteristic, the relation between d 0 (Φ f ) and Milnor numbers of the singularities of A is much blurred, in particular because Milnor number is not an invariant under contact equivalence anymore (see [HRS19] for the definition of contact equivalence and more precision about the definition of Milnor number in positive characteristic). In other words, over a field k of positive characteristic, Equation (3.1) is still valid by definition but the numbers m z cannot be interpreted as the Milnors numbers of the singularities defined by f (even if we won't need it, let us however precise that the numbers m z appeared to be related to the Milnor number of a hypersurface µ(O f ), a contact equivalent invariant defined in [HRS19, end of section 3]. We also point out that typical behaviors of singularities in positive characteristic prevent the classification of homaloidal polynomials via Dolgachev's approach in [Dol00, Lemma 3] over C, see [MHW01] and [Ngu16] for instances of such behaviors when reducing modulo p). (1) if p divides r, then m z = (r − 1) 2 + (r − 2), (2) if p does not divide r, then m z = (r − 1) 2 .
Proof. To describe m z , we first explain why it is enough to make the computation in the case that A is a near-pencil of r + 1 lines such that z is the intersection point of r lines. Once we have our local model for z, we use Theorem 2.4 to compute m z . Let A = V(f ), f = l 1 · · · l d , write z = (0 : 0 : 1) by choosing coordinates of P 2 k and label the linear polynomials l 1 , . . . , l d defining A such that l 1 , . . . , l r ∈ (x 0 , x 1 ) and l r+1 , . . . , l d ∈ (x 0 , x 1 ) c , (x 0 , x 1 ) c being the complementary of the ideal (x 0 , x 1 ).