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Register a new userOn the rationality problem for forms of moduli spaces of stable marked curves of positive genus
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Let $M_{g, n}$ (respectively, $overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n geqslant 0$ if $g geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $ eq 2$. We show that all $F$-forms of $overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 leqslant n leqslant 4$, $g = 2$ and $2 leqslant n leqslant 3$, $g = 3$ and $1 leqslant n leqslant 14$, $g = 4$ and $1 leqslant n leqslant 9$, $g = 5$ and $1 leqslant n leqslant 12$.
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We study a space of genus $g$ stable, $n$-marked tropical curves with total edge length $1$. Its rational homology is identified both with top-weight cohomology of the complex moduli space $M_{g,n}$ and with the homology of a marked version of Kontsevichs graph complex, up to a shift in degrees. We prove a contractibility criterion that applies to various large subspaces. From this we derive a description of the homotopy type of the tropical moduli space for $g = 1$, the top weight cohomology of $M_{1,n}$ as an $S_n$-representation, and additional calculations for small $(g,n)$. We also deduce a vanishing theorem for homology of marked graph complexes from vanishing of cohomology of $M_{g,n}$ in appropriate degrees, and comment on stability phenomena, or lack thereof.
We prove that Grothendieck-Witt spaces of Poincare categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubis classical hermitian and quadratic K-groups with the genuine Grothendieck-Witt groups from our joint work with Calm`es, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our solution of several conjectures in hermitian K-theory. The method of proof is abstracted from work of Galatius and Randal-Williams on cobordism categories of manifolds using the identification of the Grothendieck-Witt space of a Poincare category as the homotopy type of the associated cobordism category.
Curves of genus g which admit a map to CP1 with specified ramification profile mu over 0 and nu over infinity define a double ramification cycle DR_g(mu,nu) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR_g(mu,nu) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR_g(mu,nu) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hains formula in the compact type case. When mu and nu are both empty, the formula for double ramification cycles expresses the top Chern class lambda_g of the Hodge bundle of the moduli space of stable genus g curves as a push-forward of tautological classes supported on the divisor of nonseparating nodes. Applications to Hodge integral calculations are given.
Rapid developments in genetics and biology have led to phylogenetic methods becoming an important direction in the study of cancer and viral evolution. Although our understanding of gene biology and biochemistry has increased and is increasing at a remarkable rate, the theoretical models of genetic evolution still use the phylogenetic tree model that was introduced by Darwin in 1859 and the generalization to phylogenetic networks introduced by Grant in 1971. Darwins model uses phylogenetic trees to capture the evolutionary relationships of reproducing individuals [6]; Grants generalization to phylogenetic networks is meant to account for the phenomena of horizontal gene transfer [14]. Therefore, it is important to provide an accurate mathematical description of these models and to understand their connection with other fields of mathematics. In this article, we focus on the graph theoretical aspects of phylogenetic trees and networks and their connection to stable curves. We introduce the building blocks of evolutionary moduli spaces, the dual intersection complex of the moduli spaces of stable curves, and the categorical relationship between the phylogenetic spaces and stable curves in $overline{mathfrak{M}}_{0,n}(mathbb{C})$ and $overline{mathfrak{M}}_{0,n}(mathbb{R})$. We also show that the space of network topologies maps injectively into the boundary of $overline{mathfrak{M}}_{g,n}(mathbb{C})$.
For $4 mid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod $L$ symplectic group with coefficients in the adjoint representation.
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