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.     

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.     

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.     

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.     

Configurations in abelian categories. IV. Invariants and changing stability conditions

This is the last in a series on configurations in an abelian category A. Given a finite poset (I,?), an (I,?)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using Artin stacks. It showed well-behaved moduli stacks ObjA,MA(I,?) of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.The third introduced stability conditions(τ,T,?) on A, and showed the moduli space of τ-semistable objects in class α is a constructible subset in ObjA, so its characteristic function is a constructible function. It formed algebras , , , of constructible and stack functions on ObjA, and proved many identities in them.In this paper, if (τ,T,?) and are stability conditions on A we write in terms of the , and deduce the algebras are independent of (τ,T,?). We study invariants or Iss(I,?,κ,τ) ‘counting’ τ-semistable objects or configurations in A, which satisfy additive and multiplicative identities. We compute them completely when A=mod-KQ or A=coh(P) for P a smooth curve. We also find invariants with special properties when A=coh(P) for P a smooth surface with nef, or a Calabi-Yau 3-fold.     

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.     

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 \(Stab(D^b(\mathbb {P}^2))\) .     

Implicit algebraic geometry on categories of universal algebras

Using the notion of an implicit operation on universal algebras, we redefine basic notions of the algebraic geometry of universal algebras. The results obtained for the implicit algebraic geometry imply (as special cases) the known results on the conditional geometric and algebraic geometric comparability of algebras.     

Minkowskian geometry,convexity conditions and the parallel axiom

The problem of characterising Minkowskian spaces is an important problem of that branch of differential geometry in which spaces more general than the complete Riemann and Finsler spaces are studied axiomatically using synthetic geometric methods. The fundamental theorem in this field is the result that a Desarguesian straightG-space in which the parallel axiom holds and the spheres are convex is Minkowskian. However the question as to whether the hypothesis of the space being Desarguesian is necessary or not has remained unsolved for over forty years. It is therefore natural to investigate conditions stronger than the mere convexity of spheres. In this paper such geometric conditions derived from functions which measure the distance between lines and points on lines are studied. Besides characterising the Minkowskian spaces these investigations also bring out the interplay between the parallel axiom and the convexity and linearity conditions.     

Linear syzygies and birational combinatorics

Let F be a finite set of monomials of the same degree d ≥ 2 in a polynomial ring R = k[x ₁,…, x _n ] over an arbitrary field k. We give some necessary and/or sufficient conditions for the birationality of the ring extension k[F] ? R ^(d), where R ^(d) is the dth Veronese subring of R. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygytheoretic birationality criterion in characteristic zero obtained in [1].