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Register a new userReduction type of genus-3 curves in a special stratum of their moduli space
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We study a 3-dimensional stratum $mathcal{M}_{3,V}$ of the moduli space $mathcal{M}_3$ of curves of genus $3$ parameterizing curves $Y$ that admit a certain action of $V= C_2times C_2$. We determine the possible types of the stable reduction of these curves to characteristic different from $2$. We define invariants for $mathcal{M}_{3,V}$ and characterize the occurrence of each of the reduction types in terms of them. We also calculate the $j$-invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction of $Y$ in terms of the invariants.
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Let ${cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected etale cover ${cal M}^l$ of ${cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is almost anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let $pi_1({cal M}^l_{ol{k}})$ be the geometric algebraic fundamental group of ${cal M}^l$ and let ${Out}^*(pi_1({cal M}^l_{ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${cal M}^l$ (this is the $ast$-condition motivating the almost above). Let us denote by ${Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${cal M}^l$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({cal M}^l)cong{Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.
This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the Faber-Zagier relations on the moduli spaces of nonsingular curves and the kappa rings of the moduli spaces of curves of compact type, I present Pixtons proposal for a complete calculus of tautological classes on the moduli spaces of stable curves. Several open questions are discussed. An effort has been made to condense a great deal of mathematics into as few pages as possible with the hope that the reader will follow through to the end.
We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the etale fundamental group of a curve. Faltings $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $mathfrak{p}$ contains three irreducible components of genus $1$.
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