Showing papers in "Compositio Mathematica in 2005"
TL;DR: In this paper, the stable derived category of coherent sheaves is introduced for a noetherian scheme, and its applications include an analogue of maximal cohen-macaulay approximations, a construction of tate cohomology, and an extension of the classical Grothendieck duality.
Abstract: for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.
391 citations
TL;DR: The case p = g − 2 of the conjecture is equivalent to Noether's theorem, and the case p ≥ g − 3 to Petri's theorem (see as discussed by the authors ).
Abstract: Kl,1(C, KC) = 0, ∀l ≥ p ⇔ Cliff(C) > g − p− 2. The direction ⇒ is proved by Green and Lazarsfeld in the appendix to [4]. The case p = g − 2 of the conjecture is equivalent to Noether’s theorem, and the case p = g − 3 to Petri’s theorem (see [6]). The case p = g − 4 has been proved in any genus by Schreyer [10] and by the author [13] for g > 10. More recently, the conjecture has been studied in [11], [12], for generic curves of fixed gonality. Teixidor proves the following
182 citations
TL;DR: In this article, it was shown that if X ⊂ P r is a closed scheme in projective space whose homogeneous ideal is generated by quadrics, then a zero-dimensional or one-dimensional intersection of X with a plane of dimension p is 2-regular.
Abstract: Let X ⊂ P r be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX are linear for p steps. We show that if X satisfies N2,p then a zero-dimensional or one-dimensional intersection of X with a plane of dimension p is 2-regular. This extends a result of Green and Lazarsfeld. We give conditions when the syzygies of X restrict to the syzygies of the intersection. Many of our results also work for ideals generated by forms of higher degree. As applications, we bound the p for which some well-known projective varieties satisfy N2,p. Another application, carried out by us in a different paper, is a step
166 citations
TL;DR: In this paper, the moduli (b-) divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class was studied. And the positivity properties of the modulus (b)-divisor were investigated.
Abstract: We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.
108 citations
TL;DR: In this article, the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators is studied.
Abstract: We study bounds for the Castelnuovo–Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry ?( Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXXIV (1993), 1–48) by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyse Giusti’s proof, which provides the result in characteristic zero, giving some insight into the combinatorial properties needed in that context. For the general case, we provide a new argument which employs the Bayer–Stillman criterion for detecting regularity.
94 citations
TL;DR: In this article, the authors considered a simple complex algebraic group G with affine Grassmannian and K-theory, and derived the spectrum of homology ring with universal group-algebra centralizer of the Langlands dual group.
Abstract: For an almost simple complex algebraic group G with affine Grassmannian $\\text{Gr}_G=G(\\mathbb{C}(({\\rm t})))/G(\\mathbb{C}[[{\\rm t}]])$, we consider the equivariant homology $H^{G(\\mathbb{C}[[{\\rm t}]])}(\\text{Gr}_G)$ and K-theory $K^{G(\\mathbb{C}[[{\\rm t}]])}(\\text{Gr}_G)$. They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\\mathbb{C}[[{\\rm t}]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of $G(\\mathbb{C}[[{\\rm t}]])$-modules.
91 citations
TL;DR: In this article, it was shown that the norm of a Bethe vector in the Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point.
Abstract: We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental $sl_{r+1}$ -modules.
83 citations
TL;DR: In this paper, it was shown that a general curve of genus 14 admits a projective model D in a six-dimensional projective space, which is linked to a generalized curve C of degree 14 and genus 8 by a complete intersection of quadrics.
Abstract: We prove the unirationality of the moduli space of complex curves of genus 14. The method essentially relies on linkage of curves. In particular it is shown that a general curve of genus 14 admits a projective model D, in a six-dimensional projective space, which is linked to a general curve C of degree 14 and genus 8 by a complete intersection of quadrics. Using this property we are able to obtain, after a reasonable amount of further work, the unirationality result in the case of genus 14. Moreover, some variations of the same method, involving the Hilbert schemes of curves of very low genus, are used to obtain the same result for the known cases of genus 11, 12, 13.
78 citations
TL;DR: Weil et al. as discussed by the authors introduced an essentially new Grothendieck topology, the Weil-´ etale topology on schemes over finite fields, and the cohomology groups associated with this topology should
Abstract: We introduce an essentially new Grothendieck topology, the Weil-´ etale topology, on schemes over finite fields. The cohomology groups associated with this topology should
75 citations
TL;DR: The 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of "evenly-spaced" annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.
Abstract: We prove that near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of “evenly-spaced” annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.
72 citations
TL;DR: In this article, the Langlands correspondence is described in terms of explicit descriptions of the sets of equivalence classes of irreducible, essentially tame, supercuspidal representations of GLn(F) which are essentially tame.
Abstract: Let F be a non-Archimedean local field. Let $\\mathcal{G}_n^{\\rm et}(F)$ be the set of equivalence classes of irreducible, n-dimensional representations of the Weil group $\\mathcal{W}_F$ of F which are essentially tame. Let $\\mathcal{A}_n^{\\rm et}(F)$ be the set of equivalence classes of irreducible, essentially tame, supercuspidal representations of GLn(F). The Langlands correspondence induces a canonical bijection $\\mathcal{L}:\\mathcal{G}_n^{\\rm et}(F) \\to \\mathcal{A}_n^{\\rm et}(F)$. We continue the programme of describing this map in terms of explicit descriptions of the sets $\\mathcal{G}_n^{\\rm et}(F)$ and $\\mathcal{A}_n^{\\rm et}(F)$. These descriptions are in terms of admissible pairs $(E/F, \\xi)$, consisting of a tamely ramified field extension $E/F$ of degree n and a quasicharacter $\\xi$ of $E^\\times$ subject to certain technical conditions. If Pn(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections $P_n(F) \\cong \\mathcal{G}_n^{\\rm et}(F)$ and $P_n(F) \\cong \\mathcal{A}_n^{\\rm et}(F)$. In an earlier paper we showed that, if $\\sigma \\in \\mathcal{G}_n^{\\rm et}(F)$ corresponds to an admissible pair $(E/F,\\xi)$, then $\\mathcal{L}(\\sigma)$ corresponds to the admissible pair $(E/F,\\mu\\xi)$, for a certain tamely ramified character $\\mu$ of $E^\\times$. In this paper, we determine the character $\\mu$ when $E/F$ is totally ramified.
TL;DR: In this article, it was shown that the intersection of at least n/2 sufficiently ample general hypersurfaces in a complex abelian variety of dimension n has ample cotangent bundle.
Abstract: The aim of this article is to provide methods for constructing smooth projective complex varieties with ample cotangent bundle. We prove that the intersection of at least n/2 sufficiently ample general hypersurfaces in a complex abelian variety of dimension n has ample cotangent bundle. We also discuss analogous questions for complete intersections in the projective space. Finally, we present an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of sufficiently many smooth projective varieties with big cotangent bundle has ample cotangent bundle.
TL;DR: In this paper, the Chow ring of the complex Cayley plane is described explicitly and the most interesting Schubert varieties of the Chow rings are described and their intersection products computed.
Abstract: We give a full description of the Chow ring of the complex Cayley plane . For this, we describe explicitly the most interesting of its Schubert varieties and compute their intersection products. Translating our results into the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions Y8 introduced by the current authors in arXiv: math.AG/0306328.
TL;DR: In this article, the mixed Hodge structure on the moduli space of smooth genus 4 curves has been shown to have Poincare-Serre polynomial 1 + t2u2 + t4u4 + t5u6.
Abstract: We compute the mixed Hodge structure on the rational cohomology of the moduli space of smooth genus 4 curves. Specifically, we prove that its Poincare–Serre polynomial is 1 + t2u2 + t4u4 + t5u6. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.
TL;DR: In this article, it was shown that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces.
Abstract: We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: Every countable-dimensional real or complex locally finite Lie algebra is enlargible, i.e., it is the Lie algebra of some regular Lie group (a suitable direct limit group).
TL;DR: In this paper, the existence, structure and variance properties of a differential algebra of finite type over a field k are studied, and the main results concern the rigid dualizing complex over such an algebra A.
Abstract: A differential algebra of finite type over a field k is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated k-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when chark = 0. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to the Auslander property of the rigid dualizing complex of A.
TL;DR: In this paper, the singularities of non-confluent hypergeometric functions in complex variables are studied. And the amoebas of classical discriminantal hypersurfaces are shown to possess the minimal number of complement components.
Abstract: This paper deals with singularities of nonconfluent hypergeometric functions in several complex variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of the amoebas and the Newton polytopes of their defining polynomials. In particular, we show that the amoebas of classical discriminantal hypersurfaces are solid, that is, they possess the minimal number of complement components.
TL;DR: In this article, the index of a class of biharmonic maps derived from minimal Riemannian immersions into spheres is investigated, motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese map and the Clifford torus.
Abstract: Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese map and the Clifford torus.
TL;DR: In this article, the authors investigated divisibility properties of the traces and Hecke traces of singular moduli of a polynomial if p is prime and showed that these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields.
Abstract: We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner’s classical congruences j(z)|Up ≡ 744 (mod p )( wherep 11 and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.
TL;DR: For reductive groups over a non-archimedean field, this paper obtained results concerning the construction of simple types for the group GL(m, D), for each simple stratum of the matrix algebra.
Abstract: This work is concerned with type theory for reductive groups over a non Archimedean field. Given such a field F, and a division algebra D of finite dimension over its center F, we obtain results concerning the construction of simple types for the group GL(m, D), . More precisely, for each simple stratum of the matrix algebra M(m, D), we produce a set of β-extensions in the sense of Bushnell and Kutzko.
TL;DR: In this paper, the authors define invariants of the blow-Nash equivalence of real analytic function germs, in a similar way that the motivic zeta functions of Denef & Loeser.
Abstract: We define invariants of the blow-Nash equivalence of real analytic function germs, in a similar way that the motivic zeta functions of Denef & Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, as a generalized Euler characteristics for projective constructible arc-symmetrics sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariant, but also Nash-invariant of arc-symmetric sets. Our zeta functions enable to sketch the blow-Nash equivalence classes of Brieskorn polynomials of two variables.
TL;DR: In this paper, all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0 were explicitly computed and an explicit conjecture about what these slopes should be for weight k forms was formulated.
Abstract: We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.
TL;DR: In this article, the existence of complexified real arrangements with the same combinatorics but different embeddings in P2 has been proved and they admit conjugated equations on the ring of polynomials over Q(√5).
Abstract: We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in P2. Such a pair of arrangements has an additional property:
they admit conjugated equations on the ring of polynomials over Q(√5).
TL;DR: In this paper, the relation with similar problems on shimura varieties is explained, for a number field f and a sequence of tori hn, where f is an algebraic group.
Abstract: let g be an algebraic group, for a number field f and a sequence of tori hn. the relation with similar problems on shimura varieties is explained.
TL;DR: In this article, a model-theoretic proof of an analogue of the Manin-Mumford conjecture for semiabelian varieties in positive characteristic is presented, which reduces the general positive-characteristic Mordell-Lang problem to a question about purely inseparable points on subvarieties of semiabelians.
Abstract: We present the details of a model-theoretic proof of an analogue of the Manin–Mumford conjecture for semiabelian varieties in positive characteristic. As a by-product of the proof we reduce the general positive-characteristic Mordell–Lang problem to a question about purely inseparable points on subvarieties of semiabelian varieties.
TL;DR: For the primes l = 2, 3, 5, 7 or 13, it was shown in this article that any semi-stable abelian with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X-0(11).
Abstract: We show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over Q that have good reduction at every prime different from 1 and are semi-stable at l. We show that any semi-stable abelian variety over Q with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X-0(11). In addition, we show that for l = 2,3 and 5, there do not exist any non-zero abelian varieties over Q with good reduction outside l that acquire semi-stable reduction at l over a tamely ramified extension.
TL;DR: The core of ideals was first studied by Rees and Sally as discussed by the authors, who showed a close relationship to Lipman's adjoint ideal, and they extended the result to arbitrary dimension and more general rings.
Abstract: of I, denoted bycore(I), is defined to be the intersection of all reductions of I.The core of ideals was first studied by Rees and Sally [RS], partly due to itsconnection to the theorem of Brianc¸on and Skoda. Later, Huneke and Swanson[HuS] determined the core of integrally closed ideals in two-dimensional regularlocal rings and showed a close relationship to Lipman’s adjoint ideal. Recently,Corso, Polini and Ulrich [CPU1,2] gave explicit descriptions for the core of certainideals in Cohen-Macaulay local rings, extending the result of [HuS]. In these twopapers, several questions and conjectures were raised which provided motivationfor our work. More recently, Hyry and Smith [HyS] have shown that the core andits properties are closely related to a conjecture of Kawamata on the existence ofsections for numerically effective line bundles which are adjoint to an ample linebundle over a complex smooth algebraic variety, and they generalize the result in[HuS] to arbitrary dimension and more general rings. Nonetheless, there are manyunanswered questions on the nature of the core. One reason is that it is difficult todetermine the core and there are relatively few computed examples.Our focus in this paper is in effective computation of the core with an eye topartially answering some questions raised in [CPU1,2]. A first approach to under-standing the core was given by Rees and Sally. For an ideal Iin a local Noetherianring (R,m)having analytic spread l, one can take lgeneric generators ofIin aringofthe form R[U
TL;DR: In this article, it was shown that the subspace of Siegel cusp forms of weight $k+n$ generated by the Ikeda lifts of elliptic cusp form of weight 2 k can be characterized by simple relations among the Fourier coefficients.
Abstract: We show that for arbitrary even genus 2 n with $n\equiv {0,1}$ (mod 4) the subspace of Siegel cusp forms of weight $k+n$ generated by the Ikeda lifts of elliptic cusp forms of weight 2 k can be characterized by certain simple relations among the Fourier coefficients. These generalize the classical Maass relations in genus 2.
TL;DR: In this paper, the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT) were defined for any finite group G.
Abstract: For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the ‘quotient’ by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the ‘quotient’ by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov–Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen–Ruan orbifold Gromov–Witten invariants of of Fantechi and Gottsche.