Showing papers on "Minimal model program published in 2013"

Singularities of the minimal model program

Abstract: Preface Introduction 1. Preliminaries 2. Canonical and log canonical singularities 3. Examples 4. Adjunction and residues 5. Semi-log-canonical pairs 6. Du Bois property 7. Log centers and depth 8. Survey of further results and applications 9. Finite equivalence relations 10. Appendices References Index.

Log minimal model program for the moduli space of stable curves

Abstract: We give a geometric invariant theory (GIT) construction of the log canonical model Mg( ) of the pairs (Mg; ) for 2 (7=10 "; 7=10] for small "2 Q+. We show that Mg(7=10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; Mg(7=10 ") is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction

The essential skeleton of a degeneration of algebraic varieties

Abstract: In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let $k$ be a field of characteristic zero and let $X$ be a smooth and proper $k((t))$-variety with semi-ample canonical divisor. We prove that the essential skeleton of $X$ coincides with the skeleton of any minimal $dlt$-model and that it is a strong deformation retract of the Berkovich analytification of $X$. As an application, we show that the essential skeleton of a Calabi-Yau variety over $k((t))$ is a pseudo-manifold.

Log minimal model program for the moduli space of stable curves: the first flip

Abstract: We give a geometric invariant theory (GIT) construction of the log canonical model M ¯ g (a) of the pairs (M ¯ g ,ad) for a?(7/10�?,7/10] for small ??Q + . We show that M ¯ g (7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M ¯ g (7/10-?) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction ?:M ¯ g (7/10+?)?M ¯ g (7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction ? + :M ¯ g (7/10-?)?M ¯ g (7/10) that is the Mori flip of ? .

Uniruledness of stable base loci of adjoint linear systems via Mori theory

Abstract: We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. As a special case, we recover previous results by Takayama.

Reduction maps and minimal model theory

Abstract: We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.

Regularizing Properties of the Kähler–Ricci Flow

Abstract: These notes present a general existence result for degenerate parabolic complex Monge–Ampere equations with continuous initial data, slightly generalizing the work of Song and Tian on this topic. This result is applied to construct a Kahler–Ricci flow on varieties with log terminal singularities, in connection with the Minimal Model Program. The same circle of ideas is also used to prove a regularity result for elliptic complex Monge–Ampere equations, following Szekelyhidi–Tosatti.

Differential forms in the h-topology

Abstract: We study sheaves of differential forms and their cohomology in the h-topology. This allows to extend standard results from the case of smooth varieties to the general case. As a first application we explain the case of singularities arising in the Minimal Model Program. As a second application we consider de Rham cohomology.

On birational involutions of p 3

Abstract: We study elements of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the -fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements.

Log minimal model program for the moduli space of stable curves: The second flip

Abstract: We prove an existence theorem for good moduli spaces, and use it to construct the second flip in the log minimal model program for the moduli space of stable curves. In fact, our methods give a uniform, self-contained construction of the first three steps of the log minimal model program for the moduli spaces of stable pointed curves.

Moduli of products of stable varieties

Abstract: We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite etale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension . The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

Minimal model program with scaling and adjunction theory

Abstract: Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codimℙN(X) + 2.

On the three dimensional minimal model program in positive characteristic

Abstract: Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the flip of $f$ exists. As a consequence, we prove the existence of minimal models for any projective $\Q$-factorial terminal variety $X$ with pseudo-effective canonical divisor $K_X$.

Some remarks on the minimal model program for log canonical pairs

Abstract: We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact Kahler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-Kahler manifolds, flopping contractions, and so on, in order to clarify our results.

New outlook on the Minimal Model Program, II

Abstract: We prove that the finite generation of adjoint rings implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and termination of flips with scaling in the presence of a big boundary.

Relations in the Sarkisov program

Abstract: The Sarkisov program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If .

Stability of genus five canonical curves

Abstract: We analyze GIT stability of nets of quadrics in $\mathbb{P}^4$ up to projective equivalence. Since a general net of quadrics defines a canonically embedded smooth curve of genus five, the resulting GIT quotient gives a birational model of the moduli space of genus 5 curves. We study the geometry of the associated contraction and prove that the constructed GIT quotient is the final step of the log minimal model program for the moduli space of genus 5 curves.

The final log canonical model of $\bar{M}_6$

Abstract: 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.

On the ACC for lengths of extremal rays

Abstract: We discuss the ascending chain condition for lengths of extremal rays. We prove that the lengths of extremal rays of n-dimensional Q-factorial toric Fano varieties with Picard number one satisfy the ascending chain condition.

Log minimal model program for the moduli space of stable curves: the rst ip

Abstract: We give a geometric invariant theory (GIT) construction of the log canonical model Mg( ) of the pairs (Mg; ) for 2 (7=10 "; 7=10] for small "2 Q+. We show that Mg(7=10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; Mg(7=10 ") is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction : Mg(7=10 +")! Mg(7=10)

Around and Beyond the Canonical Class

Abstract: This survey is an invitation to recent developments in higher dimensional birational geometry.

Birational Maps in the Minimal Model Program.

Abstract: In this dissertation I face three main arguments 1) Classification of Fano-Mori contractions 2) Chern numbers on smooth threefolds 3) Pluricanonical systems

Families of canonically polarized manifolds over log Fano varieties

Abstract: Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X§is isotrivial if the divisor -(K+D) is ample. This result extends results of Viehweg-Zuo and Kebekus-Kovacs. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a Q-factorialization of X. As Q-factorializations are generally not unique, we use flops to pass from one Q-factorialization to another, proving the existence of a Q-factorialization suitable for our purposes.

Birational geometry in the study of dynamics of automorphisms and Brody/Mori/Lang hyperbolicity

Abstract: We survey our recent papers (some being joint ones) about the relation between the geometry of a compact K\"ahler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We give applications of LMMP to positivity of log canonical divisor of a Mori / Brody / Lang hyperbolic (quasi-) projective (singular) variety.

Some remarks on the minimal model program

Abstract: We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact Kahler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-Kahler manifolds, flopping contractions, and so on, in order to clarify our results.

The geometry of rational curves on some classes of varieties

Abstract: Research interests MY research interests are mainly focused on the study of birational geometry of algebraic varieties over C and especially the theory of the Minimal Model Program. More particularly, I am interested in question regarding the existence of rational curves and their interaction with the global geometry of varieties (e.g., the existence of rational curves on CY 3folds, positivity questions for log pairs), the birational geometry of Calabi Yau varieties (in particular questions regarding and consequences of the so-called Cone Conjecture) and criteria to determine whether a variety has special geometry (such as rationality, unirationality, toricness).