On the birational geometry of Fano 4-folds

We study the birational geometry of a Fano fourfold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map $f: X \dashrightarrow Y$ , where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if ρ _X ? ρ _Y  = 1, where ρ is the Picard number. We first give a characterization of non-movable prime divisors in X, when ρ _X  ≥ 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, “quasi-elementary”. The main result is that ρ _X  ≤ 11 if X has an elementary rational contraction of fiber type, and ρ _X  ≤ 18 if X has a quasi-elementary rational contraction of fiber type.     

Secant varieties and birational geometry

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties. Received: 29 November 1999; in final form: 4 September 2000 / Published online: 23 July 2001     

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when g is classical. Here, the induced orbits play an important role instead of Richardson orbits.     

Singularities of algebraic subvarieties and problems of birational geometry

We investigate 2-periodic points of a certain class of dynamical systems defined over the field of p-adic numbers. We determine the topological properties of these points and the nature of the smallest finite extension in which the periodic points reside.     

Three equivalent conjectures on the birational geometry of Fano 3-folds

I propose three equivalent conjectures on the birational geometry of Fano 3-folds. Roughly speaking, they suggest that ergodic, or chaotic, behaviour does not occur for Fano 3-folds.     

Quotient categories,stability conditions,and birational geometry

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. We prove that this category has homological dimension c. As an application, we describe the space of stability conditions on its derived category in the case c  \(=\) 1. Moreover, we describe all exact equivalences between these quotient categories in this particular case, which is closely related to classification problems in birational geometry.     

On birational Torelli theorems

Archiv der Mathematik - Let G be a simple simply-connected connected linear algebraic group over $${mathbb {C}}$$ . We prove a 2-birational Torelli theorem for the moduli space of semistable...     

The birational geometry of moduli spaces of sheaves on the projective plane

We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves \(M_H(v)\) on \(\mathbb {P}^2\) and mini-wall crossings in the stability manifold \(Stab(D^b(\mathbb {P}^2))\) .     

On certain K-equivalent birational maps

Mathematische Zeitschrift - A simple birational map is a K-equivalent birational map which is resolved by a single blowing-up. Examples of such maps include standard flops and twisted Mukai flops....     

On birational equivalence of algebraic tori

The faithful representations of quasisplit k-tori are studied. The old-standing conjecture of the rationality of stably rational tori is proved. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

On birational transformations of pairs in the complex plane

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.