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     

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 the birational geometry of

Here we are interested in birational properties of the moduli space of stable maps to the complex projective line. In particular, we determine the class of the canonical divisor in the rational Picard group and we investigate the cones of ample and effective divisors.     

Tangent algebraic subvarieties of vector fields

An algebraic commutative group is associated to any vector field on a complete algebraic variety . The group acts on and its orbits are the minimal subvarieties of which are tangent to . This group is computed in the case of a vector field on .

     

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 plane algebraic curves

We give an exposition of some of the basic results on singularities of plane algebraic curves, in terms of polynomials and formal power series.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693     

Applications of the blowing-up construction and algebraic geometry to bifurcation problems

A generalization of the Morse lemma to vector-valued functions is proved by a blowing-up argument. This is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem. Bifurcation in a one- or multi-parameter problem is guaranteed if the leading term is of even degree (it is often two) and satisfies a regularity condition. Applications are given to nonlinear eigenvalue problems and to the Hopf bifurcation.     

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.     

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 approximation of smooth submanifolds by nonsingular real algebraic subvarieties

Let M be a smooth submanifold of dimension m of a nonsingular real algebraic set X. If M can be approximated by nonsingular algebraic subsets of X, then the homology class in represented by M is algebraic. The converse, investigated in this paper, is true only in some exceptional cases.     

Coherent algebraic sheaves in real algebraic geometry

Summary In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

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Riassunto Nel presente lavoro si danno esempi di fibrati vettorialiF→X, definiti su varietà affini, regolari, compatte, connesseX che non ammettono struttura algebrica.

     

Logarithmic geometry and algebraic stacks

We construct algebraic moduli stacks of log structures and give stack-theoretic interpretations of K. Kato's notions of log flat, log smooth, and log étale morphisms. In the last section we describe the local structure of these moduli stacks in terms of toric stacks.     

Baxter's equations and algebraic geometry

Functional Analysis and Its Applications -