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We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if
is a very general smooth hypersurface of dimension
, then any dominant rational mapping
must have degree at least
. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines.
We study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian surface with itself. In both cases, we find for instance the existence of nef classes that are not pseudoeffective, answering in the negative a question raised by Grothendieck in correspondence with Mumford. We also discuss several problems and questions for further investigation.
Abstract. Multiplier ideals are associated with a complex variety and an ideal or ideal sheaf thereon, and satisfy certain vanishing theorems that have proved rich in applications, for example in local algebra. This article offers an introduction to the study of multiplier ideals, mainly adopting the geometric viewpoint.
Given a smooth complex variety X and an ideal (or ideal sheaf) a on X, one can attach to a a collection of multiplier ideals J(ac) depending on a rational weighting parameter c > 0. These ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety [Demailly 1993; Angehrn and Siu 1995; Siu 1998; Ein and Lazarsfeld 1997; 1999; Demailly 1999]. More recently they have led to the discovery of some surprising uniform results in local algebra [Ein et al. 2001; 2003; 2004]. The purpose of these lectures is to give an easy-going and gentle introduction to the algebraically-oriented local side of the theory.
Multiplier ideals can be approached (and historically emerged) from three different viewpoints. In commutative algebra they were introduced and studied by Lipman  (under the name “adjoint ideals”, which now means something else), in connection with the Briançon–Skoda theorem. On the analytic side of the field, Nadel  attached a multiplier ideal to any plurisubharmonic function, and proved a Kodaira-type vanishing theorem for them.
We give a geometric description of the loci in the arc space defined by order of contact with a given subscheme, using the resolution of singularities. This induces an identification of the valuations defined by cylinders in the arc space with divisorial valuations. In particular, we recover the description of invariants coming from the resolution of singularities in terms of arcs and jets.
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