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Register a new userPencils on surfaces with normal crossings and the Kodaira dimension of $overline{mathcal{M}}_{g,n}$
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We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $overline{mathcal{M}}_{g,n}$ is not pseudo-effective in some range, implying that $overline{mathcal{M}}_{12,6},overline{mathcal{M}}_{12,7},overline{mathcal{M}}_{13,4}$ and $overline{mathcal{M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $overline{mathcal{M}}_{12,8}$ and $overline{mathcal{M}}_{16}$. We also show that the moduli of $(4g+5)$-pointed hyperelliptic curves $mathcal{H}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the Kodaira classification for moduli of pointed hyperelliptic curves.
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We prove that the moduli spaces of curves of genus 22 and 23 are of general type. To do this, we calculate certain virtual divisor classes of small slope associated to linear series of rank 6 with quadric relations. We then develop new tropical methods for studying linear series and independence of quadrics and show that these virtual classes are represented by effective divisors.
We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $overline{mathcal A}_3$ of the moduli space ${mathcal A}_3$ of complex principally polarized abelian threefolds, and we conjecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $gge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone toroidal compactification of ${mathcal A}_g$ for any $gge 3$.
In this work we study the tau-function $Z^{1D}$ of the KP hierarchy specified by the topological 1D gravity. As an application, we present two types of algorithms to compute the orbifold Euler characteristics of $overline{mathcal M}_{g,n}$. The first is to use (fat or thin) topological recursion formulas emerging from the Virasoro constraints for $Z^{1D}$; and the second is to use a formula for the connected $n$-point functions of a KP tau-function in terms of its affine coordinates on the Sato Grassmannian. This is a sequel to an earlier work.
John Lesieutre constructed an example satisfying $kappa_sigma e kappa_ u$. This says that the proof of the inequalities in Theorems 1.3, 1.9, and Remark 3.8 in [O. Fujino, On subadditivity of the logarithmic Kodaira dimension, J. Math. Soc. Japan 69 (2017), no. 4, 1565--1581] is insufficient. We claim that some weaker inequalities still hold true and they are sufficient for various applications.
In this paper we will prove a uniformity result for the Iitaka fibration $f:X rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $kappa(X)=rm{dim}(X)-1$.
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