On the locus of curves with an odd subcanonical marked point

Abstract

We present an explicit construction of a compactification of the locus of smooth curves whose symmetric Weierstrass semigroup at a marked point is odd. The construction is an extension of Stoehr's techniques using Pinkham's equivariant deformation of monomial curves by exploring syzygies. As an application we prove the rationality of the locus for genus at most six.

On the Normal Sheaf of Gorenstein Curves

“…V ℓ is the subvector space of k r generated by the vectors v i such that d i + ℓ / ∈ S. We also have that dimT 1 s = 0, ∀ s ∈ End(S). Finally, by the very explicit and implementable method in [CF18] and [CS13], we know that the ideal of C S ⊆ P g−1 is given by suitable 1 2 (g − 3)(g − 2) quadratic and isobaric forms, when the first non zero element n 1 of S is such that 3 < n 1 g − 1 and S =< 4, 5 >, and by 1 2 (g − 3)(g − 2) quadratic and isobaric forms added to g+2 3 − 5g + 5 cubic and isobaric forms in the remanning cases. In this way, one may implement an algorithm to compute the Tjurina number τ in all this cases.…”

On nonnegatively graded Weierstrass points

“…at its unique point P = (0 : • • • : 0 : 1) at the infinity. Now we recall the construction of a compactification of M S g,1 , when S is symmetric, that was first introduced by Stoehr [St93], then improved by Contiero-Stoehr [CSt13] and generalized by Contiero-Fontes [CF18]. Let us start with a pointed canonical Gorenstein curve (C, P) whose Weierstrass semigroup S at P is symmetric.…”

“…, 2g−2 , respectively. This two avoided cases are also treated by similar techniques in [CF18], but suitable cubic forms are required to compute the ideal of the canonical Gorenstein curve C. So, making the above assumptions on the semigroup S, it follows by the Enriques-Babbage Theorem that C is nontrigonal and not isomorphic to a plane quintic. Hence the ideal of C is generated by the (g − 2)(g − 3)/2 quadratic forms F si , c.f.…”

“…4.8 Remark If τ = 1, the ideal of the canonical monomial curve C S ⊂ P 5 can not be generate by quadratic forms only, see the recent preprint by Contiero and Fontes [CF18]. In this case there are two natural compactifications of the monomial curve in A 5 : in P 5 the equations of the next lemma define the ideal of the canonical monomial curve; it is a trigonal curve whose ideal can be generated by 6 quadratic and 3 cubic equations.…”

“…In Section 3 we briefly recall a rather explicit construction of a compactification of M S g,1 when S is non-hyperelliptic and symmetric. This construction was given by Stoehr [St93], then improved by Contiero-Stoehr [CSt13] and generalized by Contiero-Fontes [CF18]. All these constructions can be viewed as a variant of Hauser's algorithm [Hau83,Hau85] to compute versal deformation spaces, see also [Stev13].…”