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A theta-characteristic on a Riemann surface $\Sigma$ is a holomorphic line bundle $L$ such that $L \otimes L \cong \omega$, i.e. a square root of the cotangent bundle. There is a moduli space (stack) …

Let me write $x_i$ and $y_i$ for a symplectic basis of cohomology of $C$, and $a_i$, $b_i$ for the linear dual basis of the first homology of $C$. It is enough to find $\delta^*(dx_i)$ and $\delta^*(d …

The notion of quasi-projective orbifold is generally accepted to contain at least the following: let $X$ be a (simply-connected) complex manifold, $G$ a group acting on $X$ by biholomorphisms, and con …

In Libgober, Anatoly S., Wood, John W. ``Differentiable structures on complete intersections. II." Singularities, Part 2 (Arcata, Calif., 1981), 123–133. the authors claim that a computer search, ba …

This is not an answer to your question, but is directly related to your remark so I thought I should mention it. I have recently proved, though I am afraid that it has not appeared yet, that moduli s …

This can be calculated from 1) the extension $\mathbb{Z} \to M_{1,1} \to SL_2(\mathbb{Z})$ and 2) the decomposition $SL_2(\mathbb{Z}) \cong \mathbb{Z}/4 *_{\mathbb{Z}/2} \mathbb{Z}/6$. The extensi …

Let $\mathfrak{M}_g$ denote the moduli stack of Riemann surfaces of genus $g$, it is a smooth complex analytic stack, and is the analytic stack underlying $\mathsf{M}_g$, the moduli stack of complex a …

This comes from the choice of the $K$-theory Thom class for complex vector bundles. Firstly, recall that $K$-theory $K^0(X)$ can be described as the group of bounded chain complexes of vector bundles …

Calculations of integral homology of $\mathcal{M}_{g, n}$ occur in Abhau, Bodigheimer, Ehrenfried (p. 3) or Godin (p. 4). Of course, in the stable range (degrees $3* \leq 2g$) the rational cohomology …

I may have miscalculated, but writing $e(\nu)$ for the Euler class of the normal bundle to such an immersion I find that $$\int_{\mathbb{CP}^3} e(\nu) c^2 = 2,$$ which appears to contradict 1). To se …

In fact the conclusion in Gorinov's paper seems to be false, see E. Looijenga, "Torelli group action on the configuration space of a surface", arXiv:2008.10556

All current proofs of Mumford's conjecture in fact prove a far stronger result, the "Strong Mumford conjecture", first formulated by Ib Madsen. This says the following (where by "moduli space" in the …

This is similar to Dan Petersen's answer, but more elementary. A fact I always mention when talking to students about matrix groups is that the diagonalisable matrices (over $\mathbb{C}$) are dense in …