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Register a new userTopology of moduli spaces of tropical curves with marked points
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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.
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This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $mathbb{P}^1$ with five marked points. The stratification structures on these moduli spaces (stacks) are investigated. We confirm Simpsons foliation conjecture of moduli space of parabolic logarithmic flat bundles for our case.
This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Browns moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove these two facts by relying on both the geometric and the algebraic aspects of the problem: the complete geometric description of the cohomology of Browns moduli spaces and the coradical filtration of cofree cooperads. This gives a conceptual proof of an identity of Bergstrom-Brown which expresses the Betti numbers of Browns moduli spaces via the inversion of a generating series. This also generalizes the Salvatore-Tauraso theorem on the nonsymmetric Lie operad.
We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $leq 6$. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for lower genera.
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 $1 leqslant n leqslant 12$.
We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Delta_g and Delta_{g,n} are simply connected, for positive g. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5.
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