Log Minimal Model Program Research Articles

The W(E6)$W(E_6)$‐invariant birational geometry of the moduli space of marked cubic surfaces

AbstractThe moduli space of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth‐century work of Cayley and Salmon. Modern interest in was restored in the 1980s by Naruki's explicit construction of a ‐equivariant smooth projective compactification of , and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification of as a natural sequence of blowups of . We describe generators for the cones of ‐invariant effective divisors and curves of both and . For Naruki's compactification , we further obtain a complete stable base locus decomposition of the ‐invariant effective cone, and as a consequence find several new ‐equivariant birational models of . Furthermore, we fully describe the log minimal model program for the KSBA compactification , with respect to the divisor , where is the boundary and is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.

Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank

AbstractLet X be a ‐factorial compact Kähler klt threefold admitting an action of a free abelian group G, which is of positive entropy and of maximal rank. After running the G‐equivariant log minimal model program, we show that such X is either rationally connected or bimeromorphic to a Q‐complex torus. In particular, we fix an issue in the proof of our previous paper [23, Theorem 1.3].

A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves

We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new moduli spaces $\bar{\mathcal{M}}_{1,n}(Q)$, which we may think of coming from an enrichment of the notion of level used to define Smyth's $m$-stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the $\bar{\mathcal{M}}_{1,n}(Q)$'s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for $\bar{\mathcal{M}}_g$.

Finiteness of log Abundant log Canonical Pairs in log Minimal Model Program with Scaling

Finiteness of log Abundant log Canonical Pairs in log Minimal Model Program with Scaling

On the log minimal model program for threefolds over imperfect fields of characteristic p>5$p>5$

We prove that many of the results of the log minimal model program hold for threefolds over fields of characteristic p > 5 $p>5$ which are not necessarily perfect. This includes the existence of flips, the cone theorem, the contraction theorem for birational extremal rays and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension 3 over algebraically closed fields, they have applications to tight closure in dimension 4.

Moduli of weighted stable elliptic surfaces and invariance of log plurigenera

Motivated by Hassett's weighted pointed stable curves, we use the log minimal model program to construct compact moduli spaces parameterizing weighted stable elliptic surfaces — elliptic fibrations with section and marked fibers each weighted between zero and one. Moreover, we show that the domain of weights admits a wall and chamber structure, describe the induced wall-crossing morphisms on the moduli spaces as the weight vector varies, and describe the surfaces that appear on the boundary of the moduli space. The main technical result is a proof of invariance of log plurigenera for slc elliptic surface pairs with arbitrary weights.

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

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.

Moduli of stable maps in genus one and logarithmic geometry, I

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to the log minimal model program for $\mathcal{M}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.

Remarks on special kinds of the relative log minimal model program

We prove $\mathbb{R}$-boundary divisor versions of results proved by Birkar or Hacon-Xu on special kinds of the relative log minimal model program.

The Coolidge–Nagata conjecture

Let E⊆P2 be a complex rational cuspidal curve contained in the projective plane. The Coolidge–Nagata conjecture asserts that E is Cremona-equivalent to a line, that is, it is mapped onto a line by some birational transformation of P2. The second author recently analyzed the log minimal model program run for the pair (X,12D), where (X,D)→(P2,E) is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in P2∖E is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

The Log Minimal Model Program for Horospherical Varieties Via Moment Polytopes

In a previous work, we described the Minimal Model Program in the family of ℚ-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (X, Δ) when X is a projective horospherical variety.

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.

Remarks on the abundance conjecture

We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical 4-folds.

Deformations of singular symplectic varieties and termination of the log minimal model program

Deformations of singular symplectic varieties and termination of the log minimal model program

An Outline of the Log Minimal Model Program for the Moduli Space of Curves

ABSTRACTWe consider the log minimal model program for the moduli space of stable curves. It is widely believed now that there is a descending sequence of critical $α$ values where the log canonical model for the moduli space of stable curves with respect to $αδ$ changes, where $δ$ denotes the divisor of singular curves. We derive a conjectural formula for the critical values in two different ways: By working out the intersection theory of the moduli space of hyperelliptic curves and by computing the GIT stability of certain curves with tails and bridges. The results give a rough outline of how the log minimal model program would proceed, predicting when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.

Birational geometry of singular moduli spaces of O'Grady type

Following Bayer and Macrì, we study the birational geometry of singular moduli spaces M of sheaves on a K3 surface X which admit symplectic resolutions. More precisely, we use the Bayer–Macrì map from the space of Bridgeland stability conditions Stab(X) to the cone of movable divisors on M to relate wall-crossing in Stab(X) to birational transformations of M. We give a complete classification of walls in Stab(X) and show that every minimal birational model of M in the sense of the log minimal model program appears as a moduli space of Bridgeland semistable objects on X. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of X and the Néron–Severi lattice of M which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of M is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.

A State Polytope Decomposition Formula

AbstractWe 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.

The final log canonical model ofℳ6

We describe the birational model of $\bar{M}_6$ given by quadric hyperplane sections of the degree 5 del Pezzo surface. In the spirit of the genus 4 case treated by Fedorchuk, we show that it is the last non-trivial space in the log minimal model program for $\bar{M}_6$. We also obtain a new upper bound for the moving slope of the moduli space.

The Coolidge–Nagata conjecture, part I

Let E⊆P2 be a complex rational cuspidal curve contained in the projective plane and let (X,D)→(P2,E) be the minimal log resolution of singularities. Applying the log Minimal Model Program to (X,12D) we prove that if E has more than two singular points or if D, which is a tree of rational curves, has more than six maximal twigs or if P2∖E is not of log general type then E is Cremona equivalent to a line, i.e. the Coolidge–Nagata conjecture for E holds. We show also that if E is not Cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in D.