Log Canonical Models Research Articles

Log canonical models for the moduli space of curves: The first divisorial contraction

In this paper, we initiate our investigation of log canonical models for ( M ¯ g , α δ ) (\overline {\mathcal {M}}_g,\alpha \delta ) as we decrease α \alpha from 1 to 0. We prove that for the first critical value α = 9 / 11 \alpha = 9/11 , the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that α = 7 / 10 \alpha = 7/10 is the next critical value, i.e., the log canonical model stays the same in the interval ( 7 / 10 , 9 / 11 ] (7/10, 9/11] . In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.

Compactifications of subvarieties of tori

We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including M0,n ⊂ M0,n (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.

Zeta Functions for Curves and Log Canonical Models

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p-adic field. When f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of f-1{0} in A2. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘q-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity. 1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)