On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors

Alina Marian; Dragos Oprea

Displaying similar documents to “On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors”

The KSBA compactification for the moduli space of degree two $K 3$ pairs

Radu Laza (2016)

Journal of the European Mathematical Society

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Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X, H)$ consisting of a degree two $K 3$ surface $X$ and an ample divisor $H$ . Specifically, we construct and describe explicitly a geometric compactification ${\bar{P}}_{2}$ for the moduli of degree two $K 3$ pairs. This compactification...

Bridgeland-stable moduli spaces for $K$ -trivial surfaces

Daniele Arcara, Aaron Bertram (2013)

Journal of the European Mathematical Society

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We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface $S$ and describe “wall-crossing behavior” for objects with the same invariants as $𝒪_{C} (H)$ when $H$ generates Pic $(S)$ and $C \in |H|$ . If, in addition, $S$ is a $K 3$ or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover...

Moduli of smoothness of functions and their derivatives

Z. Ditzian, S. Tikhonov (2007)

Studia Mathematica

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Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_{p} (T)$ and $L_{p} [- 1, 1]$ for 0 < p < ∞ using the moduli of smoothness $ω^{r} {(f, t)}_{p}$ and $ω_{φ}^{r} {(f, t)}_{p}$ respectively.

Bounds on the denominators in the canonical bundle formula

Enrica Floris (2013)

Annales de l’institut Fourier

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In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If $r$ is the Cartier index of the fibre, it was expected that $12 r$ would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in $r$ , we provide a bound $N (r)$ and an example where the smallest integer that clears the denominators of the moduli part is $N (r) / r$ . Moreover we prove that even locally the denominators...

The Kodaira dimension of the moduli space of Prym varieties

Gavril Farkas, Katharina Ludwig (2010)

Journal of the European Mathematical Society

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We study the enumerative geometry of the moduli space $ℛ_{g}$ of Prym varieties of dimension $g - 1$ . Our main result is that the compactication of $ℛ_{g}$ is of general type as soon as $g > 13$ and $g$ is different from 15. We achieve this by computing the class of two types of cycles on $ℛ_{g}$ : one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical...

Even sets of nodes on sextic surfaces

Fabrizio Catanese, Fabio Tonoli (2007)

Journal of the European Mathematical Society

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We determine the possible even sets of nodes on sextic surfaces in $ℙ^{3}$ , showing in particular that their cardinalities are exactly the numbers in the set ${24, 32, 40, 56}$ . We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, and of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification...

Relative subanalytic sheaves

Teresa Monteiro Fernandes, Luca Prelli (2014)

Fundamenta Mathematicae

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Given a real analytic manifold Y, denote by $Y_{s a}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $ℱ \mapsto ℱ^{S}$ on the category of sheaves on $Y_{s a}$ and study its properties. Roughly speaking, $ℱ^{S}$ is a sheaf on $X_{s a} \times S$ . As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.

Multiple end solutions to the Allen-Cahn equation in $ℝ^{2}$

Michał Kowalczyk, Yong Liu, Frank Pacard (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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An entire solution of the Allen-Cahn equation $Δ u = f (u)$ , where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$ , e.g. $f (u) = u (u^{2} - 1)$ , is called a $2 k$ end solution if its nodal set is asymptotic to $2 k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $- 1$ ) like the one dimensional, odd, heteroclinic solution $H$ , of $H^{''} = f (H)$ . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space...

Flexibility of surface groups in classical simple Lie groups

Inkang Kim, Pierre Pansu (2015)

Journal of the European Mathematical Society

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We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $S U (p, q)$ (resp. $S O^{*} (2 n)$ , $n$ odd) and the surface group is maximal in some $S (U (p, p) \times U (q - p)) \subset S U (p, q)$ (resp. $S O^{*} (2 n - 2) \times S O (2) \subset S O^{*} (2 n)$ ). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

Essential dimension of moduli of curves and other algebraic stacks

Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli (2011)

Journal of the European Mathematical Society

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In this paper we consider questions of the following type. Let $k$ be a base field and $K / k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$ ), what is the least transcendence degree of a field of definition of $X$ over the base field $k$ ? In other words, how many independent parameters are needed to define $X$ ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete...

Euler characteristics of moduli spaces of curves

Gilberto Bini, John Harer (2011)

Journal of the European Mathematical Society

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Let $M_{g}^{n}$ be the moduli space of $n$ -pointed Riemann surfaces of genus $g$ . Denote by $\bar{M_{g}^{n}}$ the Deligne-Mumford compactification of $M_{g}^{n}$ . In the present paper, we calculate the orbifold and the ordinary Euler characteristic of $\bar{M_{g}^{n}}$ for any $g$ and $n$ such that $n > 2 - 2 g$ .

An alternative description of the Drinfeld $p$ -adic half-plane

Stephen Kudla, Michael Rapoport (2014)

Annales de l’institut Fourier

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We show that the Deligne formal model of the Drinfeld $p$ -adic half-plane relative to a local field $F$ represents a moduli problem of polarized $O_{F}$ -modules with an action of the ring of integers in a quadratic extension $E$ of $F$ . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of ${SL}_{2} (F)$ and $SU (C) (F)$ for a two-dimensional split hermitian space $C$ for $E / F$ .

On surfaces with p $_{𝑔}$ = q = 1 and non-ruled bicanonical involution

Carlos Rito (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper classifies surfaces $S$ of general type with $p_{g} = q = 1$ having an involution $i$ such that $S / i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i .$ It is shown that $S / i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S / i$ is birational to a $K 3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^{2} = 6$ is also constructed.

Stein open subsets with analytic complements in compact complex spaces

Jing Zhang (2015)

Annales Polonici Mathematici

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Let Y be an open subset of a reduced compact complex space X such that X - Y is the support of an effective divisor D. If X is a surface and D is an effective Weil divisor, we give sufficient conditions so that Y is Stein. If X is of pure dimension d ≥ 1 and X - Y is the support of an effective Cartier divisor D, we show that Y is Stein if Y contains no compact curves, $H^{i} (Y,_{Y}) = 0$ for all i > 0, and for every point x₀ ∈ X-Y there is an n ∈ ℕ such that $Φ_{| n D |}^{- 1} (Φ_{| n D |} (x ₀)) \cap Y$ is empty or has dimension 0, where $Φ_{| n D |}$ ...

The structure of a local embedding and Chern classes of weighted blow-ups

Anca M. Mustaţǎ, Andrei Mustaţǎ (2012)

Journal of the European Mathematical Society

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For a proper local embedding between two Deligne-Mumford stacks $Y$ and $X$ , we find, under certain mild conditions, a new (possibly non-separated) Deligne-Mumford stack $X^{'}$ , with an etale, surjective and universally closed map to the target $X$ , and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to $Y$ . Moreover, a natural set of weights on the substacks of $X^{'}$ allows the construction of a universally...

Chen–Ruan Cohomology of $ℳ_{1, n}$ and ${\bar{ℳ}}_{1, n}$

Nicola Pagani (2013)

Annales de l’institut Fourier

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In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$ -pointed curves of genus $1$ . In the first part of the paper we study and describe stack theoretically the twisted sectors of $ℳ_{1, n}$ and ${\bar{ℳ}}_{1, n}$ . In the second part, we study the orbifold intersection theory of ${\bar{ℳ}}_{1, n}$ . We suggest a definition for an orbifold tautological ring in genus $1$ , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

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We continue the examination of the stable reduction and fields of moduli of $G$ -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0, p)$ , where $G$ has a $p$ -Sylow subgroup $P$ of order $p^{n}$ . Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f : Y \to ℙ^{1}$ is a three-point $G$ -Galois cover defined over $\bar{ℚ}$ , then the $n$ th higher ramification groups above $p$ for the upper numbering of the (Galois closure of...

Stratified Whitney jets and tempered ultradistributions on the subanalytic site

N. Honda, G. Morando (2011)

Bulletin de la Société Mathématique de France

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In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$ . Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$ . In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$ . Second,...

Singularities of $2 Θ$ -divisors in the jacobian

Christian Pauly, Emma Previato (2001)

Bulletin de la Société Mathématique de France

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We consider the linear system $| 2 Θ_{0} |$ of second order theta functions over the Jacobian $J C$ of a non-hyperelliptic curve $C$ . A result by J.Fay says that a divisor $D \in | 2 Θ_{0} |$ contains the origin $𝒪 \in J C$ with multiplicity $4$ if and only if $D$ contains the surface $C - C = {𝒪 (p - q) ∣ p, q \in C} \subset J C$ . In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$ , divisors containing the fourfold $C_{2} - C_{2} = {𝒪 (p + q - r - s) ∣ p, q, r, s \in C}$ , and divisors singular along $C - C$ , using...

Soluble Groups with Many Černikov Quotients

Silvana Franciosi, Francesco de Giovanni (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano i gruppi risolubili non di Černikov a quozienti propri di Černikov. Nel caso periodico tali gruppi sono tutti e soli i prodotti semidiretti $H\ltimes N$ con $N$ $p$ -gruppo abeliano elementare infinito e $H$ gruppo irriducibile di automorfismi di $N$ che sia infinito e di Černikov. Nel caso non periodico invece si riconduce tale studio a quello dei moduli a quozienti propri artiniani su un gruppo risolubile finito, e si fornisce una caratterizzazione di tali moduli.

Singularities of theta divisors and the geometry of $𝒜_{5}$

Gavril Farkas, Samuele Grushevsky, Salvati R. Manni, Alessandro Verra (2014)

Journal of the European Mathematical Society

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We study the codimension two locus $H$ in $𝒜_{g}$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $[H] \in C H^{2} (𝒜_{g})$ for every $g$ . For $g = 4$ , this turns out to be the locus of Jacobians with a vanishing theta-null. For $g = 5$ , via the Prym map we show that $H \subset 𝒜_{5}$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\bar{𝒜_{5}}$ and show that the component $\bar{N_{0}^{'}}$ of the Andreotti-Mayer...

Hyperbolic geometry and moduli of real cubic surfaces

Daniel Allcock, James A. Carlson, Domingo Toledo (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $ℳ_{0}^{ℝ}$ be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space $H^{4}$ and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in $PO (4, 1)$ . We also derive several new and several old results on the topology...

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

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We take up the study of the Brill-Noether loci $W^{r} (L, X) : = {η \in {Pic}^{0} (X) | h^{0} (L \otimes η) \geq r + 1}$ , where $X$ is a smooth projective variety of dimension $> 1$ , $L \in Pic (X)$ , and $r \geq 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^{0} (K_{D})$ , where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $> 2$ . In the $2$ -dimensional case...

Duality in $N = 2$ Susy $S U (2)$ Yang-Mills Theory : A Pedagogical Introduction to the Work of Seiberg and Witten

Adel Bilal (1997)

Recherche Coopérative sur Programme n°25

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