Weil divisors on normal surfaces (original) (raw)

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

Numerical properties of exceptional divisors of birational morphisms of smooth surfaces

Rendiconti del Circolo Matematico di Palermo Series 2

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a minimal projective normal surface with non-negative Kodaira dimension to the non minimal case.

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).

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.

Resolution except for minimal singularities

2016

The philosophy of this article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollár. Contents 21 5. Appendix. Crash course on the desingularization invariant 26 References 40

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'