Numerical properties of exceptional divisors of birational morphisms of smooth surfaces (original) (raw)

Singularities of theta divisors and the birational geometry of irregular varieties

Journal of the American Mathematical Society, 1997

The purpose of this paper is to show how the generic vanishing theorems of and can be used to settle a number of questions and conjectures raised in , Chapter 17, concerning the geometry of irregular complex projective varieties. Specifically, we focus on three sorts of results. First, we establish a well known conjecture characterizing principally polarized abelian varieties whose theta divisors are singular in codimension one. Secondly, we study the holomorphic Euler characteristic of varieties of general type having maximal Albanese dimension: we verify a conjecture of Kollár for subvarieties of abelian varieties, but show that it fails in general. Finally, we give a surprisingly simple new proof of a fundamental theorem of Kawamata [Ka] on the Albanese mapping of varieties of Kodaira dimension zero.

Equisingularity classes of birational projections of normal singularities to a plane

Advances in Mathematics, 2007

Given a birational normal extension O of a two-dimensional local regular ring (R, m), we describe all the equisingularity types of the complete m-primary ideals J in R whose blowing-up X = Bl J (R) has some point Q whose local ring O X,Q is analytically isomorphic to O. * 1 fixed a birational normal extension O of a local regular ring (R, m O ), we describe the equisingularity type of any complete m O -primary ideal J ⊂ R such that its blowing-up X = Bl J (R) has some point Q whose local ring O X,Q is analytically isomorphic to O. In this case, we will say that the surface X contains the singularity O for short, making a slight abuse of language. This is done by describing the Enriques diagram of the cluster of base points of any such ideal J: such a diagram will be called an Enriques diagram for the singularity O. Recall that an Enriques diagram is a tree together with a binary relation (proximity) representing the topological equivalence classes of clusters of points in the plane (see §1.3). Previous works by Spivakovsky and Möhring [12] describe a type of Enriques diagram that exists for any given sandwiched surface singularity (detailed in §2) and provide other types mostly in the case of cyclic quotients (see [12] 2.7) and minimal singularities (see 2.5).

Vanishing theorems and singularities in birational geometry

2013

This is a preliminary draft of monograph. It builds on lectures notes on a course that Lawrence Ein gave at the University of Catania in Summer 1998, and later again at Hong Kong University in Fall 1999, on lecture notes from the courses that Tommaso de Fernex taught at the University of Utah in Fall 2006, Spring 2010, and Spring 2012, and on lecture notes for the courses taught by Mircea Mustaţȃ in Winter and Fall 2013 at University of Michigan. This draft has been typeset using an edited version of the Springer Monograph class svmono.cls.

Divisors on some generic hypersurfaces

Journal of Differential Geometry, 1993

In this paper we consider generic hypersurfaces of degree at least 5 in P 3 and especially P 4 , and reduced, irreducible, but otherwise arbitrarily singular, divisors upon them. Our purpose is to prove that such a divisor cannot admit a desingularization having numerically effective anticanonical class. Over the past decade or so, there has been considerable interest in various questions of what might be called "generic geometry", such as the following: given a variety X which is "generic" in some sense, suppose f:Z->X is a generically finite map from a smooth variety onto some subvariety Zcl; then what can be said about the intrinsic geometry of Z? Perhaps the first, and still the most famous, instance of this problem concerns the case where X is a generic quintic hypersurface in P 4. There a conjecture of Clemens [1] is (equivalent to) the statement that Z as above must have nonnegative Kodaira dimension, i.e., cannot be birationally ruled (the usual statement of Clemens' conjecture is that X should contain only finitely many rational curves of given degree, obviously equivalent to the former statement). Coming from another direction, namely Faltings' work on the Mordell conjecture, etc., S. Lang has made a series of very general conjectures which, e.g., imply in the case of a quintic 3-fold X that Z as above cannot be an elliptic fibration, if Z = X. Along similar lines, Harris has conjectured that for X a generic surface of degree d > 5 in P 3 , Z as above cannot be a rational or elliptic curve. Harris' conjecture was recently proven by G. Xu [3], who also obtains more general bounds on the genus of Z in terms of the degree of Z. Now especially from a qualitative viewpoint, one common theme to the conjectures of Clemens and Harris stands out: that is some sort of "positivity" assertion on the canonical bundle K z. In dimension > 1 there are of course many ways to interpret such positivity, the one involved in Clemens'

Equivalent birational embeddings II: divisors

Mathematische Zeitschrift, 2012

Two divisors in P n are said to be Cremona equivalent if there is a Cremona modification sending one to the other. We produce infinitely many non equivalent divisorial embeddings of any variety of dimension at most 14. Then we study the special case of plane curves and rational surfaces. For the latter we characterise surfaces Cremona equivalent to a plane.

Resolution of Singularities of Pairs

2016

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at a ∈ X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f : X → X such that the transformed pair (X, D) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollár. Contents 1. Introduction 2 2. Characterization of semi-snc points 4 3. Basic notions and structure of the proof 6 4. The Hilbert-Samuel function and semi-simple normal crossings 12 5. Algorithm for the main theorem 20 6. The case of more than 2 components 23 7. The case of two components 31 8. The non-reduced case 37 9. Functoriality 39 References 40

A Theory of Divisors for Algebraic Curves

2007

The purpose of this paper is twofold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in [6], where, following Severi, we introduced the notion of the "branch" of an algebraic curve. Secondly, we use results from [6], in order to develop a refined theory of g r n on an algebraic curve. This refinement depends critically on replacing the notion of a point with that of a "branch". We are then able to construct a theory of divisors, generalising the corresponding theory in the special case when the algebraic curve is non-singular, which is birationally invariant.