Birational Morphism Research Articles

Decomposition of Log Crepant Birational Morphisms between Log Terminal Surfaces

We prove that every log crepant birational morphism between log terminal surfaces can be decomposed into log flopping type divisorial contraction morphisms and log blowdowns. Repeating these two kinds of contractions we reach a minimal log minimal surface from any log minimal surface.

All toric local complete intersection singularities admit projective crepant resolutions

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.

On Macaulayfication of Noetherian schemes

The Macaulayfication of a Noetherian scheme X is a birational proper morphism from a Cohen-Macaulay scheme to X. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension 0 or 1. In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic 0. Our method, however, to construct a Macaulayfication is independent of the characteristic.

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let Xhol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on Xhol (X normal algebraic surface) by algebraic vector bundles.

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction.

Germs of arcs on singular algebraic varieties and motivic integration

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of Oesterle.

Poisson structures and birational morphisms associated with bundles on elliptic curves

In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study birational morphisms between projective spaces appearing as such moduli spaces.

Birational Morphisms of Rational Surfaces

A fairly complete characterization on anticanonical rational surfaces of the divisor classes whose sections define birational morphisms is given. Most results are proven over an algebraically closed field of arbitrary characteristic.

Weak Hironaka Theorem

The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, $\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$. Then there exist a smooth projective variety $M$, a strict normal crossings divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D = R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also use a multidimensional version of G.Belyi argument which allows us to simplify the shape of a ramification divisor. By induction on the dimension of $X$ the problem is reduced to resolving toroidal singularities. This process however is too crude and does not permit any control over the structure of the birational map $f$. A different proof of the same theorem was found independently by D. Abramovich and A.J. de Jong. The approach is similar in both proofs but they seem to be rather different in details.

A Regularity Theorem on Birational Morphisms

A Regularity Theorem on Birational Morphisms

Very ample linear systems on abelian varieties

Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$ is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the following: (1) If $d \ge g+2$, then $\phi_L\colon X \to {\bf P}^{d-1}$ defines a birational morphism onto its image. (2) If $d > 2^g$, then $L$ is very ample. We show the latter by checking it on a suitable rank-$(g-1)$-degeneration.

Minimal Order Morphisms

An order is a one-dimensional, residually finite, Noetherian domain. A finite birational morphism ƒ: A → B between two orders can be decomposed by means of minimal morphisms: inert, decomposed, or ramified. We determine a least upper bound for the number of minimal morphisms appearing in a decomposition of ƒ. This number is reached in the canonical decomposition ƒ = ƒ 1∘ƒ 2∘ƒ 3 where ƒ 1: A″ → B (resp. ƒ 2: A′ → A″, ƒ 3: A → A′) is composed only of inert morphisms (resp. decomposed, ramified). The orders A′ and A″ are unique and are respectively the seminormalization and the quasinormalization of A in B. The canonical decomposition always exists. Though there is a decomposition with a minimal number of minimal order morphisms, we can only give a lower bound for this number. All this applies especially to the orders of algebraic number fields and an example is given.

Factorization of birational morphisms of regular schemes

Factorization of birational morphisms of regular schemes

Generalized parabolic sheaves on an integral projective curve

We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spacesM(X) of GPS of certain type onX. IfX is obtained by blowing up finitely many nodes inY then we show that there is a surjective birational morphism from M(X) to M (Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curveY.

A remark on the geography of surfaces with birational canonical morphisms

A remark on the geography of surfaces with birational canonical morphisms

The Uniform Bound Problem for Local Birational Nonsingular Morphisms

It is known that any factorization of a local birational morphism $f:\operatorname {Spec}\;S \to \operatorname {Spec}\;R$ of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length. This fact generalizes the classical Local Factorization Theorem of Zariski and Abhyankar, which states that there is a unique such factorization, that given by quadratic transformations, in the surface case. A much stronger generalization is given here, namely, that there exists a uniform bound on the lengths of all such factorizations, provided that $R$ is excellent. This bound is explicitly calculated for some concrete extensions and examples are given to show that this is the strongest generalization possible in some sense.

Homeomorphism versus isomorphism for varieties

It is well known that a morphism onto a weakly normal algebraic variety that is both birational and a universal homemorphism is an isomorphism of varieties. in this note, the author shows that a birational morphism onto a weakly normal algebraic variety X with a continuous inverse is necessarily an isomorphism of varieties if and only if X had no one-dimensional components. Thus the characterization of the weak normalization of a variety X as the largest variety that is birationally homeomorphic to X is only valid in the absence of one-dimensional components. All varieties are over a fixed algebraically closed field.

On unique and almost unique factorization of complete ideals II

The theory of complete ideals was begun by Zariski in 1938 [Z 1], and further developed in Appendix 5 of [ZS]. Zariski developed a theory of factorization of complete ideals in regular two-dimensional local rings. In this paper we develop the theory of factorization of complete ideals in complete normal two-dimensional local rings. Let (R, m) be a complete normal two-dimensional local ring, and let re(R) be the semi-group of complete m-primary ideals of R. We show how the divisor class group of R, CI(R), is related to factorization in re(R). We generalize results obtained in [C2] where it is assumed that R/m is algebraically closed. Assume that R is any regular local ring of dimension 2. Zariski proved that any complete ideal I of R has unique factorization into products of simple complete ideals. The distinct simple complete ideals occuring in the factorization of I are in 1 1 correspondence with the irreducible components of the exceptional divisor of the blowing up of I. This shows that any proper birational morphism X ~spec(R) with X normal can be decomposed into a sequence of blowups with irreducible exceptional divisors. Now assume that R is a complete normal local domain of dimension 2. It is natural to ask if the condition that m(R) has unique factorization is equivalent to R being a UFD. We show (Theorem 5) that if re(R) has unique factorization then R is a UFD. We also give an example (Example 1) to show that the converse does not hold. However, Lipman has shown in [L] that if R is a UFD, and if R/m is algebraically closed, then re(R) has unique factorization. Recall that R is a UFD if and only if CI(R)=0. G6hner [G] introduced the notion of a semigroup being semi-factorial. This definition is reproduced in definition 3 of this paper. We prove G6hner's conjecture (2 3 of Theorem 4) that re(R) is semifactorial if and only if CI(R) is a torsion group. As a corollary (Corollary to Theorem 4) we obtain that if Rim is algebraically closed of characteristic zero, then re(R) is semi-factorial if and only if R has a rational singularity. Another corollary is that CI(R) is torsion if and only if R satisfies the condition (N) of Muhly and Sakuma [-MS] (1r of Theorem 4). R satisfies condition (N) if any proper birational morphism X ~ spec(R) factors as a sequence of blowups of height 2 ideals with irreducible exceptional divisors.

The uniform bound problem for local birational nonsingular morphisms

It is known that any factorization of a local birational morphism f : Spec S → Spec R f:\operatorname {Spec}\;S \to \operatorname {Spec}\;R of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length. This fact generalizes the classical Local Factorization Theorem of Zariski and Abhyankar, which states that there is a unique such factorization, that given by quadratic transformations, in the surface case. A much stronger generalization is given here, namely, that there exists a uniform bound on the lengths of all such factorizations, provided that R R is excellent. This bound is explicitly calculated for some concrete extensions and examples are given to show that this is the strongest generalization possible in some sense.

Sur l'espace de modules des faisceaux semi stables de rang 2, de classes de Chern (0,3) sur ${\Bbb P}_2$

The moduli space of semi-stable sheaves of rank-2 and Chern classes (0,3) on the projective plane ℙ 2 , denoted by M=M(0,3), is a smooth irreducible projective variety of dimension 9. In M, the points which come from sheaves which are not locally free give a hypersurface ∂M. In this article we show that all complete surfaces in M must meet the boundary ∂M; in other words, there does not exist a family of vector bundles which is parametrized by a complete surface in M. The essential point of the proof is the construction of a birational morphism Φ:M→, Gr from M to the grassmannian Gr of nets of conics in ℙ 2 ; this allows us to identify M with a blow-up of Gr along a surface. This gives us a precise description of cohomology algebra of M and we use this to determine the fundamental class of a complete surface in M which does not meet ∂M. We then show that this value of the fundamental class can not arise.