Showing papers on "Minimal model program published in 2003"

Notes on toric varieties from Mori theoretic viewpoint

Abstract: The main purpose of this notes is to supplement the paper by Reid: De- composition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. As an application, a generalization of Fujita's c onjecture for singular toric varieties is obtained. We also prove that every toric variety has a small projective toric Q-factorialization. 0. Introduction. The main purpose of this notes is to supplement the paper by Reid (Re): Decomposition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. This is an answer to (Ma, Remark-Question 10-3-6) for toric varieties, which is an easy exercise once we understand (Re). As a corollary, we obtain a strong version of Fujita's conjecture for singular toric varieties. Related topics are (Ft), (Ka), (La) and (Mu, Section 4). We will work, throughout this paper, over an algebraically closed field k of arbitrary characteristic.

Local simple connectedness of resolutions of log-terminal singularities

Abstract: We study fundamental groups related with log-terminal singularities, and show that fundamental groups are preserved by a resolution of singularities. As corollaries, we show that fundamental groups are invariant under various fundamental operations in the Minimal Model Program, for example, by contractions of extremal rays, by flips, by pluricanonical morphisms of minimal varieties of general type.

Introduction to the toric Mori theory

Abstract: The main purpose of this paper is to give a simple and non-combinatorial proof of the toric Mori theory. Here, the toric Mori theory means the (log) Minimal Model Program (MMP, for short) for toric varieties. We minimize the arguments on fans and their decompositions. We recommend this paper to the following people: (A) those who are uncomfortable with manipulating fans and their decompositions, (B) those who are familiar with toric geometry but not with the MMP. People in the category (A) will be relieved from tedious combinatorial arguments in several problems. Those in the category (B) will discover the potential of the toric Mori theory. As applications, we treat the Zariski decomposition on toric varieties and the partial resolutions of non-degenerate hypersurface singularities. By these applications, the reader will learn to use the toric Mori theory.

Stringy zeta functions for ℚ-Gorenstein varieties

Abstract: The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the minimal model program, we introduce very general stringy invariants, associated to "almost all" singularities, more precisely, to all singularities that are not strictly log canonical. They specialize to the invariants of Batyrev when the singularity is log terminal. For example, the simplest form of our stringy zeta function is, in general, a rational function in one variable, but it is just a constant (Batyrev's stringy Euler number) in the log terminal case.

Stringy zeta functions for Q-Gorenstein varieties

Abstract: The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities, introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties, and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the Minimal Model Program, we introduce very general stringy invariants, associated to 'almost all' singularities, more precisely to all singularities which are not strictly log canonical. They specialize to the invariants of Batyrev when the singularity is log terminal. For example the simplest form of our stringy zeta function is in general a rational function in one variable, but it is just a constant (Batyrev's stringy Euler number) in the log terminal case.

Compact moduli of hyperplane arrangements

Abstract: The minimal model program suggests a compactification of the moduli space of hyperplane arrangements which is a moduli space of stable pairs. Here, a stable pair consists of a scheme X which is a degeneration of projective space and a divisor D=D_1+..+D_n on X which is a limit of hyperplane arrangements. For example, in the 1-dimensional case, the stable pairs are stable curves of genus 0 with n marked points. Kapranov has defined an alternative compactification using his Chow quotient construction, which may be described fairly explicitly. We prove that these two compactifications coincide. We deduce a description of all stable pairs.

Equivariant completions of toric contraction morphisms

Abstract: We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-$\mathbb Q$-factorial toric varieties. So, our theory seems to be quite different from Reid's original combinatorial toric Mori theory. We also explain various examples of non-$\mathbb Q$-factorial contractions, which imply that the $\mathbb Q$-factoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Program.

An application of the canonical bundle formula

Abstract: We prove a part of Shokurov's conjecture on characterization of toric varieties modulo the minimal model program and adjunction conjecture.

Topological Euler numbers in a semi-stable degeneration of surfaces

Abstract: The object of this paper is to study topological Euler numbers in a semi-stable degeneration of surfaces by using the semi-stable minimal model program. As its application, we find some restrictions of singularities in a semi-stable degeneration of surfaces with general fiber a minimal $\kappa = 0$ surface.