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3.0

Jun 30, 2018
06/18

by
Daniel Plaumann; Mihai Putinar

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We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

Topics: Complex Variables, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1410.5935

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4.0

Jun 29, 2018
06/18

by
Laure Flapan

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We consider algebraic surfaces, recently constructed by Schreieder, which are smooth models of the quotient of the self-product of a complex hyperelliptic curve by a $\mathbb{Z}/3^c\mathbb{Z}$-action. We show that these surfaces are elliptic modular surfaces attached to a particular subgroup of index $6\cdot 3^c$ in $SL(2,\mathbb{Z})$. In particular, this implies that Schreieder's surfaces are extremal elliptic surfaces, meaning they have maximal Picard rank and Mordell-Weil rank $0$. In fact,...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1603.05613

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9.0

Jun 30, 2018
06/18

by
Yang-Hui He; Cyril Matti; Chuang Sun

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The so-called Scattering Equations which govern the kinematics of the scattering of massless particles in arbitrary dimensions have recently been cast into a system of homogeneous polynomials. We study these as affine and projective geometries which we call Scattering Varieties by analyzing such properties as Hilbert series, Euler characteristic and singularities. Interestingly, we find structures such as affine Calabi-Yau threefolds as well as singular K3 and Fano varieties.

Topics: High Energy Physics - Theory, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1403.6833

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10.0

Jun 27, 2018
06/18

by
Young-Hoon Kiem; In-Kyun Kim; Hwayoung Lee; Kyoung-Seog Lee

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We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective varieties and raise many interesting questions.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1503.00125

3
3.0

Jun 30, 2018
06/18

by
Joseph Tenini

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In 2002, Alexeev provided a modular interpretation for the toroidal compactification of $A_g$ for the 2nd Voronoi fan. In this paper we show that pairs $(X,\Theta)$ in the boundary of this moduli space are semi-log canonical, the analog of log canonical in the non-normal setting. This extends the result of Koll${\'a}$r stating that principally polarized abelian pairs $(X,\Theta)$ are log canonical.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1401.0516

3
3.0

Jun 28, 2018
06/18

by
Jaiung Jun

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We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani

Topics: Group Theory, Number Theory, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1510.02979

3
3.0

Jun 29, 2018
06/18

by
Bhargav Bhatt

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Andr\'e recently gave a beautiful proof of Hochster's direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where "discriminants" acquire all $p$-power roots. In this paper, we explain a quicker proof of Hochster's conjecture that circumvents the perfectoid Abhyankar lemma; instead,...

Topics: Commutative Algebra, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1608.08882

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4.0

Jun 29, 2018
06/18

by
Chunyi Li

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The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on P2 has a principle connected component Stab^\dag(P2). We show that Stab^\dag(P2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab^\dag (P2) and show that Stab^\dag(P2) is contractible.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1611.02087

4
4.0

Jun 29, 2018
06/18

by
Ke Chen; Xin Lu; Kang Zuo

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In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $g\geq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.

Topics: Number Theory, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1611.08477

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7.0

Jun 30, 2018
06/18

by
Volker Braun; David R. Morrison

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We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various $SU(5)\times U(1)^n$ and $E_6$ models are presented as examples. To each genus-one fibration one can associate a $\tau$-function on the base as well as an $SL(2,\mathbb{Z})$ representation...

Topics: High Energy Physics - Theory, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1401.7844

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Jun 27, 2018
06/18

by
Alexey Kalugin

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In present paper we develop categorical formalism of Verdier duality for diagrams of topoi. We use this approach to construct Grothendieck six operations formalism.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1505.06922

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6.0

Jun 28, 2018
06/18

by
Alessandro Verra

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Let S be a general complex Nikulin surface of genus 8, a geometric construction of S is given as follows. Consider a smooth 3-fold linear section T of the Grassmannian G(1,4) and the Hilbert scheme of rational normal sextic curves of T. In it consider the special family of sextics A which are also contained in the congruence of bisecant lines to a rational normal quartic curve of P^4. We show that S is biregular to a quadratic section of T containing a sextic A. In particular A admits a...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1509.03364

7
7.0

Jun 29, 2018
06/18

by
Jean-Louis Colliot-Thélène

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Let K be the function field of a curve over the complex field. Let X be a homogeneous space of a semisimple linear algebraic group. Strong approximation holds for X outside any finite nonempty set of places of K. Strong approximation fails for tori over K.

Topics: Group Theory, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1604.03464

3
3.0

Jun 30, 2018
06/18

by
Ana Cristina López Martín; Carlos Tejero Prieto

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We study derived equivalences of Abelian varieties in terms of their associated symplectic data. For simple Abelian varieties over an algebraically closed field of characteristic zero we prove that the natural correspondence introduced by Orlov, which maps equivalences to symplectic isomorphisms, is surjective.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1702.00232

3
3.0

Jun 29, 2018
06/18

by
Eugene Gorsky; Andrei Neguţ; Jacob Rasmussen

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We construct a categorification of the maximal commutative subalgebra of the type $A$ Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by...

Topics: Representation Theory, Geometric Topology, Algebraic Geometry, Quantum Algebra, Mathematics

Source: http://arxiv.org/abs/1608.07308

4
4.0

Jun 30, 2018
06/18

by
Alan Adolphson; Steven Sperber

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The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is $\leq n$, the zeta function is trivial mod $p$ and the Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all $d$. We also describe the...

Topics: Algebraic Geometry, Number Theory, Mathematics

Source: http://arxiv.org/abs/1701.04509

2
2.0

Jun 30, 2018
06/18

by
Javier Fernández de Bobadilla; Jawad Snoussi; Mark Spivakovsky

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We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1405.6760

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3.0

Jun 30, 2018
06/18

by
Daniel Litt

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We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.

Topics: Mathematics, Number Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1405.7380

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2.0

Jun 30, 2018
06/18

by
Indranil Biswas; Sean Lawton; Daniel Ramras

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We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.

Topics: Mathematics, Algebraic Topology, Representation Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1412.4389

2
2.0

Jun 30, 2018
06/18

by
Hiroshi Iritani; Etienne Mann; Thierry Mignon

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We give an interpretation of quantum Serre of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E^\vee over X, and (3) the quantum D-module of a submanifold Z\subset X cut out by a regular section of E. When E is the anticanonical line...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1412.4523

2
2.0

Jun 30, 2018
06/18

by
Qilin Yang

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In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact K\"ahler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.

Topics: Complex Variables, Mathematics, Differential Geometry, Algebraic Geometry

Source: http://arxiv.org/abs/1412.7869

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3.0

Jun 30, 2018
06/18

by
Ciro Ciliberto; Maria Angelica Cueto; Massimiliano Mella; Kristian Ranestad; Piotr Zwiernik

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In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1403.1814

2
2.0

Jun 30, 2018
06/18

by
Gaël Cousin

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We discuss two results about projective representations of fundamental groups of quasiprojective varieties. The first is a realization result which, under a nonresonance assumption, allows to realize such representations as monodromy representations of flat projective logarithmic connections. The second is a lifting result: any representation as above, after restriction to a Zariski open set and finite pull-back, can be lifted to a linear representation.

Topics: Complex Variables, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1403.6954

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3.0

Jun 30, 2018
06/18

by
Megumi Harada; Tatsuya Horiguchi; Mikiya Masuda

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Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring H^*_S(Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal J generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group S \cong \mathbb{C}^* is a certain subgroup of a maximal torus T...

Topics: Mathematics, Algebraic Topology, Algebraic Geometry

Source: http://arxiv.org/abs/1405.1785

2
2.0

Jun 30, 2018
06/18

by
Efthymios Sofos

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We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.

Topics: Mathematics, Number Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1402.0303

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3.0

Jun 30, 2018
06/18

by
Alexander Braverman; David Kazhdan

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In this note we formulate and prove a version of Cartan decomposition for holomorphic loop groups, similar to Cartan decomposition for $p$-adic loop groups, discussed proved by Garland (and later by the authors by geometric mathods). The main technical tool that we use is the (well-known) interpretation of twisted conjugacy classes in the holomorphic loop group in terms of principal holomorphic bundles on an elliptic curve.

Topics: Mathematics, Representation Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1402.1369

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4.0

Jun 30, 2018
06/18

by
Matthew Woolf

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In this paper, we study rational sections of the relative Picard variety of a linear system on a smooth projective variety. Specifically, we prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard variety come from restrictions of line bundles on the variety.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1402.2128

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2.0

Jun 30, 2018
06/18

by
Karl Fredrickson

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Using an inclusion of one reflexive polytope into another is a well-known strategy for connecting the moduli spaces of two Calabi-Yau families. In this paper we look at the question of when an inclusion of reflexive polytopes determines a torically-defined extremal transition between smooth Calabi-Yau hypersurface families. We show this is always possible for reflexive polytopes in dimensions two and three. However, in dimension four and higher, obstructions can occur. This leads to a smooth...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1402.4785

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3.0

Jun 30, 2018
06/18

by
Niels Lubbes

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We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1402.2454

2
2.0

Jun 30, 2018
06/18

by
Hoang Thanh Hoai; Ichiro Shimada

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Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q+1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1402.3372

2
2.0

Jun 30, 2018
06/18

by
Fabrizio Catanese; Appendix by Jonathan Wahl

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The paper answers a question by Jonathan Wahl,giving examples of regular surfaces S (so their canonical ring is a Gorenstein graded ring) having the following properties: 1) their canonical divisor K_S = rL is a positive multiple of an ample divisor L 2) the graded ring R := R (X,L ) associated to L is not Cohen-Macaulay. In the appendix Wahl shows how these examples lead to the existence of Cohen-Macaulay singularities with K_X Q -Cartier which are not Q -Gorenstein, since their index one...

Topics: Mathematics, Commutative Algebra, Algebraic Geometry

Source: http://arxiv.org/abs/1402.3815

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3.0

Jun 30, 2018
06/18

by
Andrea Cattaneo

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Let $B$ be a smooth projective surface, and $\mathcal{L}$ an ample line bundle on $B$. The aim of this parer is to study the families of elliptic Calabi--Yau threefolds sitting in the bundle $\mathbb{P}(\mathcal{L}^a \oplus \mathcal{L}^b \oplus \mathcal{O}_B)$ as anticanonical divisors. We will show that the number of such families is finite.

Topics: Mathematics, High Energy Physics - Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1402.4383

2
2.0

Jun 30, 2018
06/18

by
Jean-Louis Colliot-Thélène; Alena Pirutka

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Inspir\'es par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\bf P}^4_{\mathbb C}$ qui ne sont pas stablement rationnelles, plus pr\'ecis\'ement dont le groupe de Chow de degr\'e z\'ero n'est pas universellement \'egal \`a $\mathbb Z$. --- There are (many) smooth quartic hypersurfaces in ${\bf P}^4_{\mathbb C}$ which are not stably rational. More precisely, their degree zero Chow group is not universally equal to $\mathbb Z$. The proof uses a...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1402.4153

2
2.0

Jun 30, 2018
06/18

by
Damian Rössler

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We give a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1412.7303

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5.0

Jun 30, 2018
06/18

by
Katharina Heinrich

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In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves $C$ together with a finite map $i: C \to \mathbb{P}^3$ that is generically a closed immersion and such that $C$ has Hilbert polynomial $p(t)=3t+1$ with respect to $i$. We show that CM is irreducible, smooth and birational to one component of the Hilbert scheme of twisted cubics.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1403.6441

2
2.0

Jun 30, 2018
06/18

by
Patrick Clarke

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We study what we call quasi-spline sheaves over locally Noetherian schemes. This is done with the intention of considering splines from the point of view of moduli theory. In other words, we study the way in which certain objects that arise in the theory of splines can be made to depend on parameters. In addition to quasi-spline sheaves, we treat ideal difference-conditions, and individual quasi- splines. Under certain hypotheses each of these types of objects admits a fine moduli scheme. The...

Topics: Mathematics, Numerical Analysis, Algebraic Geometry

Source: http://arxiv.org/abs/1403.6625

2
2.0

Jun 30, 2018
06/18

by
Sylvain Brochard

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We study in this article the dual of a (strictly) commutative group stack $G$ and give some applications. Using the Picard functor and the Picard stack of $G$, we first give some sufficient conditions for $G$ to be dualizable. Then, for an algebraic stack $X$ with suitable assumptions, we define an Albanese morphism $a_X : X\to A^1(X)$ where $A^1(X)$ is a torsor under the dual commutative group stack $A^0(X)$ of $Pic_{X/S}$. We prove that $a_X$ satisfies a natural universal property. We give...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1404.0285

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2.0

Jun 30, 2018
06/18

by
Gufang Zhao

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A localization theorem for the cyclotomic rational Cherednik algebra $H_c=H_c((\mathbb{Z}/l)^n\rtimes \mathfrak{S}_n)$ over a field of positive characteristic has been proved by Bezrukavnikov, Finkelberg and Ginzburg. Localizations with different parameters give different $t$-structures on the derived category of coherent sheaves on the Hilbert scheme of points on a surface. In this short note, we concentrate on the comparison between different $t$-structures coming from different...

Topics: Mathematics, Representation Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1405.1898

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3.0

Jun 30, 2018
06/18

by
Charles Siegel

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We study the moduli space ${V}_4\mathcal{M}_{g}$ of Klein four covers of genus $g$ curves and its natural compactification. This requires the construction of a related space which has a choice of basis for the Klein four group. This space has the interesting property that the two components intersect along a component of the boundary. Further, we carry out a detailed analysis of the boundary, determining components, degrees of the components over their images in $\overline{\mathcal{M}_g}$, and...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1407.3530

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5.0

Jun 30, 2018
06/18

by
Yifan Chen

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Let $S$ be a smooth minimal complex surface of general type with $p_g=0$ and $K^2=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^2=7$ is isomorphic to $\mathbb{Z}_2^2$ or $\mathbb{Z}_2 \times \mathbb{Z}_4$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_4$.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1408.2254

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2.0

Jun 30, 2018
06/18

by
Magdalena Lampa-Baczyńska; Grzegorz Malara

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The purpose of this note is to study containment relations and asymptotic invariants for ideals of fixed codimension skeletons (simplicial ideals) determined by arrangements of $n + 1$ general hyperplanes in the $n-$dimensional projective space over an arbitrary field.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1408.2472

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3.0

Jun 29, 2018
06/18

by
Alfio Ragusa; Giuseppe Zappalà

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We investigate the standard generalized Gorenstein algebras of homological dimension three, giving a structure theorem for their resolutions. Moreover in many cases we are able to give a complete description of their graded Betti numbers.

Topics: Commutative Algebra, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1612.02472

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Jun 27, 2018
06/18

by
David Kazhdan; Yakov Varshavsky

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In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of...

Topics: Algebraic Geometry, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1504.07859

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3.0

Jun 30, 2018
06/18

by
Changho Keem; Yun-Hwan Kim

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We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this note, we show that any non-empty $\mathcal{H}_{g+2,g,4}$ is irreducible without any restriction on the genus $g$. Our result augments the irreducibility result obtained earlier by Hristo Iliev(2006), in which several low genus $g\le 10$ cases have been left...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1703.07516

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3.0

Jun 30, 2018
06/18

by
Ronan J. Conlon; Rafe Mazzeo; Frédéric Rochon

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We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi-Yau metrics admit a polyhomogeneous expansion at infinity, a result that we extend to asymptotically conical Calabi-Yau metrics as well. We then study the moduli space of Calabi-Yau deformations that fix the...

Topics: Mathematics, Analysis of PDEs, Differential Geometry, Algebraic Geometry

Source: http://arxiv.org/abs/1408.6562

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Jun 30, 2018
06/18

by
Ciro Ciliberto; Flaminio Flamini; Mikhail Zaidenberg

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In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d$ at least 5 in $\mathbb{P}^3$ (the cases $d \leqslant 4$ are well known). We introduce the set $Gaps(d)$ of all non-negative integers which are not realized as geometric genera of irreducible curves on $S$. We prove that $Gaps(d)$ is finite and, in particular, that $Gaps(5)= \{0,1,2\}$. The set $Gaps(d)$ is the union of finitely many disjoint and...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1409.2758

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Jun 30, 2018
06/18

by
Gueo Grantcharov; Mehdi Lejmi; Misha Verbitsky

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A hypercomplex manifold $M$ is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it is locally obtained as a Hessian of a function averaged with quaternions. HKT metric is a natural analogue of a...

Topics: Mathematics, Differential Geometry, Algebraic Geometry

Source: http://arxiv.org/abs/1409.3280

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Jun 30, 2018
06/18

by
Goulwen Fichou; Ronan Quarez; Jean-Philippe Monnier

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We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these functions correspond to rational functions which become regular after some blowings-up, we work on the plane where it suffices to blow-up points and then we can count the number of stages of blowings-up necessary. In the latest parts of the paper, we investigate the...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1409.8223

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Jun 30, 2018
06/18

by
Tom Coates; Hiroshi Iritani; Yunfeng Jiang

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Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y,...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1410.0024

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Jun 30, 2018
06/18

by
Marian Aprodu; Gavril Farkas; Angela Ortega

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We prove two results. First, we establish that the normal bundle of any smooth curve of genus 7 having maximal Clifford index is stable. Note that 7 is the smallest genus for which such a result could possibly hold. We then show that rank four Lazarsfeld-Mukai vector bundles on a curve that lies on a general K3 surface are stable. Both results have consequences for Mercat's conjecture on higher rank vector bundles on generic curves.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1410.0857