[Paper Review] On the locus of curves with an odd subcanonical marked point

André Contiero, Aislan Leal Fontes|arXiv (Cornell University)|Apr 25, 2018

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper constructs an explicit compactification of the moduli space of smooth curves with an odd subcanonical marked point using Pinkham's equivariant deformation theory and syzygy analysis of canonical ideals. The key contribution is proving rationality of the moduli space for genus ≤ 6, with explicit equations provided for genus 5 and 6 cases via deformation of monomial Gorenstein curves.

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'sequivariant deformation of monomial curves by exploring syzygies. As an application we prove the rationality of the locus for genus at most six.

Motivation & Objective

• To construct an explicit compactification of the moduli space of smooth curves with an odd subcanonical marked point.
• To extend Stoehr’s techniques beyond the usual restrictions on symmetric semigroups, particularly avoiding hyperelliptic and odd Weierstrass semigroups.
• To investigate the global geometry of moduli spaces $\mathcal{M}_{g,1}^{\mathcal{S}}$ for odd symmetric semigroups $\mathcal{S}$ using syzygy analysis.
• To prove rationality of $\overline{\mathcal{M}_{g,1}^{\mathcal{S}}}$ for $g \leq 6$ by constructing explicit equations in weighted projective space.

Proposed method

• Deforms the canonical ideal of a monomial Gorenstein curve associated with an odd symmetric semigroup $\mathcal{S}$, using Pinkham’s equivariant deformation theory.
• Analyzes syzygies of the canonical ideal to control the deformation space and construct the compactification as a closed subscheme of $\mathbb{P}(T^{1,-}(\mathbf{k}[\mathcal{S}]))$.
• Uses the negatively graded part of the first cotangent complex $T^{1,-}(\mathbf{k}[\mathcal{S}])$ as the ambient space for the compactified moduli space.
• Applies Buschweitz’s description of the obstruction space $T^2$ to detect non-trivial obstructions in genus 5 and 6 cases.
• Derives explicit isobaric polynomial equations for $\overline{\mathcal{M}_{6,1}^{\mathcal{S}}}$ in terms of deformation parameters.
• Parametrizes the moduli space locally via rational functions after setting $a_5 = 1$, confirming rationality.

Experimental results

Research questions

• RQ1 Can an explicit compactification be constructed for the moduli space of curves with an odd subcanonical marked point, beyond the scope of Stoehr’s original framework?
• RQ2 How do syzygies of the canonical ideal influence the deformation theory of monomial Gorenstein curves with odd symmetric semigroups?
• RQ3 What is the geometric structure of $\overline{\mathcal{M}_{g,1}^{\mathcal{S}}}$ for odd symmetric semigroups $\mathcal{S}$ of genus $g \leq 6$?
• RQ4 Are the moduli spaces $\mathcal{M}_{g,1}^{\mathcal{S}}$ rational for $g \leq 6$, particularly when obstructions to smoothing exist?
• RQ5 What is the role of the cotangent complex $T^1$ and $T^2$ in determining the rationality and structure of the compactified moduli space?

Key findings

• The moduli space $\overline{\mathcal{M}_{6,1}^{\mathcal{S}}}$ is explicitly described as the zero locus of five isobaric polynomials $\vartheta_{13}, \vartheta_{15}, \vartheta_{16}, \vartheta_{17}, \vartheta_{19}$ in 15 variables.
• The local parametrization of $\overline{\mathcal{M}_{6,1}^{\mathcal{S}}}$ is rational, with $b_{10}, b_{12}, b_8$ expressed rationally in terms of $a_4, a_5=1, a_7, a_8, a_9, b_3, b_4, b_5, b_6, b_8, b_{10}, b_{11}, b_{12}$.
• For genus 5 with $\mathcal{S} = \langle 5,6,7,8\rangle$, the degree $-9$ part of $T^2$ has dimension 1, indicating non-trivial obstructions to smoothing.
• For genus 6 with $\mathcal{S} = \langle 6,7,8,9,10\rangle$, the degree $-13$ part of $T^2$ has dimension 1, confirming non-vanishing obstruction space.
• The compactification $\overline{\mathcal{M}_{g,1}^{\mathcal{S}}}$ is a closed subscheme of the weighted projective space $\mathbb{P}(T^{1,-}(\mathbf{k}[\mathcal{S}]))$, constructed via syzygy analysis.
• The rationality of $\mathcal{M}_{g,1}^{\mathcal{S}}$ for $g \leq 6$ is confirmed via explicit rational parametrization, despite non-trivial $T^2$.