ON THE BIRATIONALITY OF THE HESSIAN MAPS OF QUARTIC CURVES AND CUBIC SURFACES

We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof of the similar property for cubic surfaces, which is already known by the work of these two authors.


Introduction
Let S(n) be the graded polynomial ring in x 0 , x 1 , . . ., x n with complex coefficients.For a degree d homogeneous polynomial f ∈ S(n) d , the associated Hessian polynomial hess(f ) = det ∂ 2 f ∂x i ∂x j is either 0, or a homogeneous polynomial in S(n) 3(d−2) .Since for a generic choice of f one has hess(f ) = 0, we get a rational Hessian map h d,n : P(S(n) d ) P(S(n) 3(d−2) ), h d,n (f ) = hess(f ).
This rational map is studied in detail in [1], where the following result on binary forms is proved.induces a birational morphism onto its image if and only if d ≥ 5.
The authors of [1] propose the following very interesting conjecture.
They prove this conjecture for (d, n) = (3, 3) and show that h 3,2 is not a birational morphism onto its image.In this note we show that this conjecture holds for ternary forms of degree 4, namely we prove the following result.
We also give a new proof, completely different from that given in [1], of the following result, already mentioned above.
Our new approach is the following.We note first that there is a natural SL n+1 (C)action on any projective space P(S(n) k ) and that the Hessian map h d,n is equivariant with respect to these actions, a fact already used in [1].Then we start with the Fermat polynomial ), but the computations look too complicated for us.
To simplify the notation, we set in the sequel x = x 0 , y = x 1 , z = x 2 and t = x 3 .Computations with Singular [2] played a key role in our results.

The case of cubic surfaces in P 3
We start with the proof of Proposition 1.4, since it is much easier and shorter than the proof of Theorem 1.3.
We consider the Fermat surface of degree 3, namely (2.1) The corresponding SL 4 (C)-orbit, denoted by SL 4 (C) • f 0 has a tangent space at f 0 , spanned by all the monomials in x, y, z which are divisible by at least one of x 2 , y 2 , z 2 and t 2 , see for instance [3].It follows that the 4-dimensional affine space T inside P(S(3) 3 ) given by (2.2) with a, b, c, d ∈ C form a transversal to the orbit SL 4 (C) • f 0 in the point given by f 0 .It follows that the union of the orbits SL 4 (C) • f with f ∈ T is a constructible subset in P(S(3) 3 ), containing a non-empty Zariski open set.The Hessian polynomial hess(f ) has the following form where the sum is over all multi-indices α = (α 1 , α 2 , α 3 , α 4 ) with |α| = α 1 + α 2 + α 3 + α 4 = 4, and h α is a polynomial in a, b, c, d for any α.
It follows that, on the Zariski open subset of the transversal T given by abcd = 0, one has the following formulas It follows that the hessian map h 3,3 is injective on a non-empty Zariski open subset of P(S(3) 3 ), and the corresponding inverse mapping is given by rational functions.This completes the proof of Proposition 1.4.

The case of quartic curves in P 2
We consider now the Fermat curve of degree 4, namely (3.1) The corresponding SL 3 (C)-orbit has a tangent space at f 0 , spanned by all the monomials in x, y, z which are divisible by at least one of x 3 , y 3 and z 3 , see for instance is a bijection onto its image when restricted to a non-empty Zariski open subset of C 6 .Moreover, the inverse mapping is given by rational functions.
We list first some of the polynomials h α , as computed using Singular.