15 research outputs found
Mori Dream Spaces
This article is based on the 7th Takagi Lectures that the author delivered at the University of Tokyo on November 21-23, 2009.We explore the circle of ideas connecting finite generation of the Cox ring, Mori dream spaces and invariant theory
The cone of pseudo-effective divisors of log varieties after Batyrev
In these notes we investigate the cone of nef curves of projective varieties,
which is the dual cone to the cone of pseudo-effective divisors. We prove a
structure theorem for the cone of nef curves of projective -factorial klt pairs of arbitrary dimension from the point of view of the
Minimal Model Program. This is a generalization of Batyrev's structure theorem
for the cone of nef curves of projective terminal threefolds.Comment: 15 pages. v2: Completely rewritten paper. Structure theorem for the
cone of nef curves proved in arbitrary dimension using results of Birkar,
Cascini, Hacon and McKernan. To appear in Mathematische Zeitschrif
Termination of (many) 4-dimensional log flips
We prove that any sequence of 4-dimensional log flips that begins with a klt
pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor,
terminates. This implies termination of flips that begin with a log Fano pair
and termination of flips in a relative birational setting. We also prove
termination of directed flips with big K+D. As a consequence, we prove
existence of minimal models of 4-dimensional dlt pairs of general type,
existence of 5-dimensional log flips, and rationality of Kodaira energy in
dimension 4.Comment: 13 pages; a minor change in the proof of Thm.4.
Characterization of the 4-canonical birationality of algebraic threefolds
In this article we present a 3-dimensional analogue of a well-known theorem
of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of
surfaces of general type. Let be a projective minimal 3-fold of general
type with -factorial terminal singularities and the geometric genus
. We show that the 4-canonical map is {\it not}
birational onto its image if and only if is birationally fibred by a family
of irreducible curves of geometric genus 2 with
where is a general irreducible member in .Comment: 25 pages, to appear in Mathematische Zeitschrif
Systems of Hess-Appel'rot type
We construct higher-dimensional generalizations of the classical
Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter
leading to an algebro-geometric integration of this new class of systems, which
is closely related to the integration of the Lagrange bitop performed by us
recently and uses Mumford relation for theta divisors of double unramified
coverings. Based on the basic properties satisfied by such a class of systems
related to bi-Poisson structure, quasi-homogeneity, and conditions on the
Kowalevski exponents, we suggest an axiomatic approach leading to what we call
the "class of systems of Hess-Appel'rot type".Comment: 40 pages. Comm. Math. Phys. (to appear
Weakly--exceptional quotient singularities
A singularity is said to be weakly--exceptional if it has a unique purely log
terminal blow up. In dimension , V. Shokurov proved that weakly--exceptional
quotient singularities are exactly those of types , , ,
. This paper classifies the weakly--exceptional quotient singularities
in dimensions and