Tommaso de Fernex

The bad locus of a free linear system ℒ on a normal complex projective variety X is defined as the set B( ℒ ) ⊆ X of points that are not contained in any irreducible and reduced member of ℒ. In this paper we provide a geometric description of such locus in terms of the morphism defined by ℒ. In particular, assume that dim X ≥ 2 and ℒ is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(ℒ) is empty unless X is a surface. Then we prove that, when the latter occurs, B(ℒ) is not empty if and only if ℒ defines a morphism onto a two dimensional cone, in which case B(ℒ) is the inverse image of the vertex of the cone.

In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as $K$-theory, index theory, motivic integration and elliptic genera.

The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that the dimension of X is at least 2, and that L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if L defines a morphism onto a two dimensional cone, in which case B(L) is the inverse image of the vertex of the cone.

This paper contains a new proof of the classification of prime order elements of Bir(ℙ²) up to conjugation. The first results on this topic can be traced back to classic works by Bertini and Kantor, among others. The innovation introduced by this paper consists of explicit geometric constructions of these Cremona transformations and the parameterization of their conjugacy classes. The methods employed here are inspired to [4], and rely on the reduction of the problem to classifying prime order automorphisms of rational surfaces. This classification is completed by combining equivariant Mori theory to the analysis of the action on anticanonical rings, which leads to characterize the cases that occur by explicit equations (see [28] for a different approach). Analogous constructions in higher dimensions are also discussed.

This paper contains a new proof of the classification of elements of prime order in the Cremona group Bir(P^2), up to conjugation. In addition, we give explicit geometric constructions of these Cremona transformations, and provide a parameterization of their conjugacy classes. Analogous constructions in higher dimensions are also discussed.

We use intersection theory, degeneration techniques and jet schemes to study log canonical thresholds. Our first result gives a lower bound for the log canonical threshold of a pair in terms of the log canonical threshold of the image by a suitable smooth morphism. This in turn is based on an inequality relating the log canonical threshold and the Samuel multiplicity, generalizing our previous result from math.AG/0205171. We then give a lower bound for the log canonical threshold of an affine scheme defined by homogeneous equations of the same degree in terms of the dimension of the non log terminal locus (this part supersedes math.AG/0105113). As an application of our results, we prove the birational superrigidity of every smooth hypersurface of degree N in P^N, if 4\leq N\leq 12.