Smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$

International audience We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.

Second flip in the Hassett–Keel program: a local description

This is the first of three papers in which we give a moduli interpretation of the second flip in the log minimal model program for $\overline{M}_{g}$, replacing the locus of curves with a genus $2$ Weierstrass tail by a locus of curves with a ramphoid cusp. In this paper, for $\unicode[STIX]{x1D6FC}\in (2/3-\unicode[STIX]{x1D716},2/3+\unicode[STIX]{x1D716})$, we introduce new $\unicode[STIX]{x1D6FC}$-stability conditions for curves and prove that they are deformation open. This yields algebraic stacks $\overline{{\mathcal{M}}}_{g}(\unicode[STIX]{x1D6FC})$ related by open immersions $\overline{{\mathcal{M}}}_{g}(2/3+\unicode[STIX]{x1D716}){\hookrightarrow}\overline{{\mathcal{M}}}_{g}(2/3){\hookleftarrow}\overline{{\mathcal{M}}}_{g}(2/3-\unicode[STIX]{x1D716})$. We prove that around a curve $C$ corresponding to a closed point in $\overline{{\mathcal{M}}}_{g}(2/3)$, these open immersions are locally modeled by variation of geometric invariant theory for the action of $\text{Aut}(C)$ on the first-order deformation space of $C$.

THE LMMP FOR LOG CANONICAL 3-FOLDS IN CHARACTERISTIC

We prove that one can run the log minimal model program for log canonical 3-fold pairs in characteristic $p>5$. In particular, we prove the cone theorem, contraction theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.

Singularities with 𝔾 m -action and\break the log minimal model program for ℳ¯ g

We give a precise formulation of the modularity principle for the log canonical models

A State Polytope Decomposition Formula

We give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.