Showing papers in "Michigan Mathematical Journal in 2000"
TL;DR: In this paper, the authors characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients in affine space.
Abstract: The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the chamber decomposition of the cone of divisors in Mori theory is naturally identified with the decomposition of the G-ample cone from geometric invariant theory. In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction can be read off from the chamber decomposition.
441 citations
TL;DR: In this article, a smooth complex quasi-projective variety is considered and a Q-divisor is associated to the multiplier ideal sheaf J (D) = J (X,D) ⊆ OX, whose zeros are supported on the locus at which the pair of X,D fails to have log-terminal singularities.
Abstract: Let X be a smooth complex quasi-projective variety, and let D be an effective Q-divisor on X. One can associate to D its multiplier ideal sheaf J (D) = J (X,D) ⊆ OX , whose zeroes are supported on the locus at which the pair (X,D) fails to have log-terminal singularities. It is useful to think of J (D) as reflecting in a somewhat subtle way the singularities of D: the “worse” the singularities, the smaller the ideal. These ideals and their variants have come to play an increasingly important role in higher dimensional geometry, largely because of their strong vanishing properties. Among the papers in which they figure prominently, we might mention for instance [30], [4], [33], [2], [13], [34], [19], [14] and [8]. See [6] for a survey.
214 citations
TL;DR: The relationship between the pencils of formal flat connections and homology of a new series of pointed stable curves of genus zero was established in this article, where the role of operadic algebras was taken over by the representations of the appropriately twisted homology algebra of this union.
Abstract: It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series $\bar{L}_n$ of pointed stable curves of genus zero Whereas $\bar{M}_{0,n+1}$ parametrizes trees of $\bold{P}^1$'s with pairwise distinct nonsingular marked points, $\bar{L}_n$ parametrizes strings of $\bold{P}^1$'s stabilized by marked points of two types The union of all $\bar{L}_n$'s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union
193 citations
TL;DR: A complete list of all possible ADE-types of singular fibers of complex elliptic K3 surfaces and the torsion parts of their MordellWeil groups is given in this paper.
Abstract: We make a complete list of all possible ADE-types of singular fibers of complex elliptic K3 surfaces and the torsion parts of their MordellWeil groups.
121 citations
TL;DR: The notion of globally F-regularity was introduced in this article for algebraic groups with strong vanishing properties, including the vanishing of higher cohomology groups for any numerically effective line bundle (as discussed above for Fano varieties).
Abstract: H (X, (L ⊗ ω−1) ⊗ ω) vanishes whenL ⊗ ω−1 is ample, and hence vanishing holds in particular whenever L is numerically effective andω−1 is ample. In this paper, a class of algebraic varieties is introduced, the class of globally F-regular varieties.Globally F-regular varieties have strong vanishing properties, including the vanishing of the higher cohomology groups for any numerically effective line bundle (as discussed above for Fano varieties). Indeed, the class of globally F-regular varieties of characteristic 0 is shown to include Fano varieties, so the vanishing just described is recovered. A nice feature of the class of globally F-regular varieties is that it is preserved under the operation of forming certain (and conjecturally: any) GIT quotients by linearly reductive groups. Globally F-regular varieties are closely related to Frobenius split varieties [MRn]. Both Frobenius splitting and global F-regularity are notions defined using the Frobenius morphism in characteristic p; by reduction to characteristic p, both Frobenius splitting and global F-regularity make sense in characteristic 0 as well. As explained within, global F-regularity turns out to be a stable version of the notion of Frobenius split along a divisor that has arisen in the Indian school of algebraic groups [MRn; RR; R1; R2]. However, the definition of global F-regularity is based on the theory of tight closure introduced by Hochster and Huneke in [HH1]: roughly speaking, a projective algebraic variety is globally F-regular if it has a coordinate ring in which all ideals are tightly closed. The original motivation for this work was a question of Allen Knutson in his study [Kn] of torus actions in symplectic geometry: Let G be a semi-simple complex algebraic group with fixed Borel subgroup B and maximal torusT ⊂ B. Consider the geometric invariant theory (GIT) quotient X of the homogeneous spaceG/B with respect to some choice of linearization of the natural left action
101 citations
TL;DR: In this paper, a new perspective on the intersection theory of the moduli space of curves was described, which encompasses advances from both classical degeneracy studies and topological gravity, and the main new results were computations of basic Hodge integral series in A∗(Mg) encoding the canonical evaluations of κg−2−iλi.
Abstract: 0.1. Overview. Let Mg be the moduli space of Deligne–Mumford stable curves of genus g ≥ 2. The study of the Chow ring of the moduli space of curves was initiated by Mumford in [Mu]. In the past two decades, many remarkable properties of these intersection rings have been discovered. Our first goal in this paper is to describe a new perspective on the intersection theory of the moduli space of curves that encompasses advances from both classical degeneracy studies and topological gravity. This approach is developed in Sections 0.2–0.7. The main new results of the paper are computations of basic Hodge integral series in A∗(Mg) encoding the canonical evaluations of κg−2−iλi . The motivation for the study of these tautological elements and the series results are given in Section 0.8. The body of the paper contains the Hodge integral derivations.
98 citations
TL;DR: In this paper, the authors classify the family of spacelike maximal surfaces in Lorentz-Minkowski 3-space L 3 which are foliated by pieces of circles.
Abstract: We classify the family of spacelike maximal surfaces in Lorentz-Minkowski 3-space L 3 which are foliated by pieces of circles. This space contains a curve of singly periodic maximal surfaces R that play the same role as Riemann’s minimal examples in E. As a consequence, we prove that maximal spacelike annuli in L 3 bounded by two circles in parallel spacelike planes are pieces of either a catenoid or a surface in R.
82 citations
TL;DR: In this paper, a K-theoretic version of the Witten-Dijkgraaf-Verlinde-Ver linde equation is introduced to express the associativity constraint of the "quantum multiplication" operation on K(X).
Abstract: 0. Introduction. Quantum cohomology theory can be described in general words as intersection theory in spaces of holomorphic curves in a given Kahler or almost Kahler manifold X. By quantum K-theory we may similarly understand a study of complex vector bundles over the spaces of holomorphic curves in X. In these notes, we will introduce a K-theoretic version of the Witten-Dijkgraaf-Verlinde-Verlinde equation which expresses the associativity constraint of the “quantum multiplication” operation on K(X). Intersection indices of cohomology theory, ∫
75 citations
TL;DR: In this paper, the authors review some known results about homaloidal polynomials and also classify them in the cases when F has no multiple factors and either n = 3 or n = 4 and F is the product of linear polynomial.
Abstract: is a birational map. We shall call F with such property a homaloidal polynomial. In this paper we review some known results about homaloidal polynomials and also classify them in the cases when F has no multiple factors and either n = 3 or n = 4 and F is the product of linear polynomials. I am grateful to Pavel Etingof, David Kazhdan, and Alexander Polishchuk for bringing to my attention the problem of classification of homaloidal polynomials and for various conversations on this matter. Also I thank Hal Schenck for making useful comments on my paper.
69 citations
58 citations
TL;DR: In this article, the notion of a test exponent for tight closure was introduced, and its relationship with the problem of showing that tight closure commutes with local cohomology was explored.
Abstract: We introduce the notion of a test exponent for tight closure, and explore its relationship with the problem of showing that tight closure commutes with localization, a longstanding open question Roughly speaking, test exponents exist if and only if tight closure commutes with localization: mild conditions on the ring are needed to prove this We give other, independent, conditions that are necessary and sufficient for tight closure to commute with localization in the general case, in terms of behavior of certain associated primes and behavior of exponents needed to annihilate local cohomology While certain related conditions (the ones given here are weaker) were previously known to be sufficient, these are the first conditions of this type that are actually equivalent The difficult calculation of§4 uses associativity of multiplicities and many other tools to show that sufficient conditions for localization to commute with tight closure can be given in which asymptotic statements about lengths of modules defined using the iterates of the Frobenius endomorphism replace the finiteness conditions on sets of primes introduced in §3 The result is local and requires special conditions on the rings: one is that countable prime avoidance holds This is not a very restrictive condition however: it suffices, for example, for the ring to contain an uncountable field Countable prime avoidance also holds in any complete local ring But we also need the existence of a strong test ideal (see
TL;DR: In this article, it was shown that if a strictly pseudoconvex real-analytic hypersurface 0 ⊂ C is called spherical at a pointp ∈ 0 if there exists a germ of a biholomorphic map at p from 0 to ∂B, then it is biholomorphically equivalent to spherical at any point.
Abstract: We will usually identify the germζ f with one of its representatives—that is, a map f : U → C defined in a small neighborhood U 3 ζ and satisfyingf(U ∩ 0) ⊂ 0 ′. Several authors have studied this problem. Alexander [A] generalized Poincaré’s theorem to higher dimensions in 1974. A year later, Pinchuk [P1] proved that any germ of a biholomorphic mapping from a connected strictly pseudoconvex real-analytic hypersurface 0 ⊂ C to ∂B extends analytically along any path on0 as a locally biholomorphic map with the inclusion f(0) ⊂ ∂B. Recall that a strictly pseudoconvex real-analytic hypersurface 0 ⊂ C is called sphericalat a pointp ∈0 if there exists a germ of a biholomorphic map at p from 0 to ∂B. It follows from [P1] that, if a connected strictly pseudoconvex hypersurface is spherical at a point, then it is spherical at any point. Pinchuk’s result clearly holds if, in the target space, ∂B is replaced by an arbitrary simply connected compact strictly pseudoconvex spherical hypersurface 0 ′. Indeed, if0 ′ is spherical then a germ of a biholomorphic mapping g : 0 ′ → ∂B extends along any path on0 ′. Since0 ′ is simply connected, g extends to a global mapping from 0 ′ to ∂B. But then0 ′ is biholomorphically equivalent to ∂B. If 0 ′ is not simply connected, the result is no longer true. In fact, Burns and Shnider [BS] constructed some examples of compact and spherical but not simply connected hypersurfaces inC. For any such hypersurface 0 ′ ⊂ C, there exists a germ of a biholomorphic mappingf : ∂B → 0 ′ that does not extend holomorphically along some paths on ∂B.
TL;DR: In this paper, it was shown that the Betti numbers of a smooth projective divisor are determined by its cohomology class if the divisors are ample or if the first Betti number of X is zero.
Abstract: We have three main results. First, we show that a smooth complex projective variety which contains three disjoint codimension-one subvarieties in the same homology class must be the union of a whole one-parameter family of disjoint codimension-one subsets. More precisely, the variety maps onto a smooth curve with the three given divisors as fibers, possibly multiple fibers (Theorem 2.1). The beauty of this statement is that it fails completely if we have only two disjoint divisors in a homology class, as we will explain. The result seems to be new already for curves on a surface. The key to the proof is the Albanese map. We need Theorem 2.1 for our investigation of a question proposed by Fulton, as part of the study of degeneracy loci. Suppose we have a line bundle on a smooth projective variety which has a holomorphic section whose divisor of zeros is smooth. Can we compute the Betti numbers of this divisor in terms of the given variety and the first Chern class of the line bundle? Equivalently, can we compute the Betti numbers of any smooth divisor in a smooth projective variety X in terms of its cohomology class in H2(X,Z)? The point is that the Betti numbers (and Hodge numbers) of a smooth divisor are determined by its cohomology class if the divisor is ample or if the first Betti number of X is zero (see section 4). We want to know if the Betti numbers of a smooth divisor are determined by its cohomology class without these restrictions. The answer is no. In fact, there is a variety which contains two homologous smooth divisors, one of which is connected while the other is not connected. Fortunately, we can show that this is a rare phenomenon: if a variety contains a connected smooth divisor which is homologous to a non-connected smooth divisor, then it has a surjective morphism to a curve with some multiple fibers, and the two divisors are both unions of fibers. This is our second main result, Theorem 5.1. We also give an example of two connected smooth divisors which are homologous but have different Betti numbers. Conjecture 6.1, suggested by this example, asserts that two connected smooth divisors in a smooth complex projective variety X which are homologous should have cyclic etale coverings which are deformation equivalent to each other. The third main result of this paper, Theorem 6.3, is that this conjecture holds, in a slightly weaker form (allowing deformations into positive characteristic), under the strange assumption that the Picard variety of X is isogenous to a product of elliptic curves. The statement in general would follow from a well-known open problem in the arithmetic theory of abelian varieties, Conjecture 6.2: for any abelian variety A over a number field F , there are infinitely many primes p of the ring of integers oF such that the finite group A(oF /p) has order prime to the characteristic of the field oF /p.
TL;DR: In this article, it was shown that if f is a mapping of the plane onto itself that is uniformly continuous with modulus of continuity Ω(r) which is o( √ r) as r → 0 and f is also co-uniformly continuous, then f = P ◦ h where h is a homeomorphism and P is a complex polynomial.
Abstract: It is shown that if f is a mapping of the plane onto itself that is uniformly continuous with modulus of continuity Ω(r) which is o( √ r) as r → 0 and f is also co-uniformly continuous then f = P ◦ h where h is a homeomorphism of the plane and P is a complex polynomial. The same conclusion holds also under other assumptions on the moduli of uniform and co-uniform continuity. However, we also present an example showing that this does not hold for all uniform quotient mappings: There is a mapping of the plane onto itself whose moduli of uniform and co-uniform continuity are both of power type but it maps an interval to zero. We also discuss uniform quotient mappings of the plane onto the line. Subject classification: 54E15, 57N05.
TL;DR: In this paper, the authors generalize the techniques used by Nadirashvili to obtain new examples of complete minimal surfaces inside a ball in R^3, with the conformal structure of an annulus.
Abstract: In 1996, Nadirashvili used Runge's theorem to produce a complete minimal disc inside a ball in R^3. In this paper we generalize the techniques used by Nadirashvili to obtain new examples of complete minimal surfaces inside a ball in R^3, with the conformal structure of an annulus.
TL;DR: In this article, it was shown that there is a rational curve C in U such that the fundamental group of C surjects onto the fundamental groups of U. As a consequence, they obtained new proofs of the theorems of Harbater and Colliot-Threshold on Galois covers and torsors over the p-adic line.
Abstract: Let U be an open subset of a unirational variety. We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U. As a consequence we obtain new proofs of the theorems of Harbater and Colliot-Th\'el\`ene on Galois covers and torsors over the p-adic line. We also obtain examples of pencils of curves of genus <14 whose monodromy group is the full Teichm\"uller group.
TL;DR: In this paper, the Julia set J is shown to be a hyperbolic set for polynomial automorphisms and the complete information about the Hausdorff dimensions of J+ and J− is already contained in the Julia sets J itself.
Abstract: Let g be a polynomial automorphism of C2. In a similar way as is done for polynomials inC, we denote byK± the set of points inC2 with bounded forward/ backward orbit under g. We writeJ ± = ∂K± andJ = J+ ∩ J−. We refer toJ ± as the positive/negative Julia set and to J as the Julia set of g. The setJ ± is unbounded, closed, and connected, while J is compact (see [BS2; BS3; FM; HO] for more details). The purpose of the main part of this paper is to show that, under the assumption thatg is a hyperbolic mapping (i.e., the Julia set J is a hyperbolic set for g), the complete information about the Hausdorff dimensions of J+ andJ− is already contained in the Julia set J itself. In particular, the results of Theorem 4.1–4.4 can be summarized by the following result.
TL;DR: In this article, an algorithm for computing the degree of the closure of the linear orbit of an arbitrary plane curve with irreducible singularities is presented. But the main tool is an intersection-theoretic study of the projective normal cone of a plane curve determined by the curve in the projected projective space.
Abstract: The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work.
TL;DR: In this article, it was shown that if X is a locally nite n-dimensional polyhedron such that X Q admits a Z-compacti cation, then X X I also admits Z Z.
Abstract: We prove that if X is a locally nite n-dimensional polyhedron such that X Q admits a Z-compacti cation, then X I also admits a Z-compacti cation. Our argument relies on an extension of Dierker's Lemma [6], [2] which says that if P p and Q are locally nite polyhedra, p 3 and c : P ! Q is a PL surjection with contractible pointinverses, then Q I collapses to P . See Proposition 1.5 for details, including control on the collapse. In the last section, we give an example of a uniformly contractible manifold with bounded geometry which does not satisfy Chapman-Siebenmann's tameness condition at in nity and which therefore does not admit a Z-compacti cation.
TL;DR: In this paper, the proof of the theorem is presented, which is more elementary than the special cases that were treated in [10] and [11] and resembles proofs in [6] and '12].
Abstract: Necessary preliminaries are dealt with in Section 2. After some preparations, wegive in Section 3 the proof of the theorem. Surprisingly, it is more elementary thanthe special cases that were treated in [10]; part of it resembles proofs in [6] and [9].Acknowledgment. Part of the work on this paper was completed while I vis-ited Universite Paul Sabatier at Toulouse. I wish to thank this institution for itshospitality and the members of the Laboratoire Emile Picard for interesting andstimulating discussions.
TL;DR: In this paper, the authors present a general method for constructing real solutions to enumerative geometry problems, which gives lower bounds on the maximum number of real solutions, and apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real.
Abstract: We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these other enumerative problems.
TL;DR: This paper showed that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z) generated by symplectic transvections.
Abstract: Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, the number of Gamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number of simple reflections that appear in a reduced decomposition for u or v and F_2 is the two-element field.