Kummer Variety Research Articles

Some equations for the universal Kummer variety

We give a method to find quartic equations for Kummer varieties and we give some explicit examples. From these equations for g g -dimensional Kummer varieties one obtains equations for the moduli space of g + 1 g+1 -dimensional Kummer varieties. These again define modular forms which vanish on the period matrices of Riemann surfaces. The modular forms that we find for g = 5 g=5 appear to be new and of lower weight than known before.

On the Coble quartic and Fourier–Jacobi expansion of theta relations

In [Q. Ren, S. Sam, G. Schrader and B. Sturmfels, The universal Kummer threefold, Experiment Math.22(3) (2013) 327–362], the authors conjectured equations for the universal Kummer variety in genus 3 case. Although, most of these equations are obtained from the Fourier–Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic, cf. [A. Coble, Algebraic Geometry and Theta Functions, American Mathematical Society Colloquium Publications, Vol. 10 (American Mathematical Society, 1929)] was obtained differently, cf. [S. Grushevsky and R. Salvati Manni, On Coble's quartic, preprint (2012), arXiv:1212.1895] too. The aim of this paper is to show that Coble's quartic can be obtained as Fourier–Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in [Experiment Math.22(3) (2013) 327–362].

A Proof of the Kawamata–Morrison Cone Conjecture for Holomorphic Symplectic Varieties ofK3[n]or Generalized Kummer Deformation Type

We prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.

Explicit constructions for genus 3 Jacobians

Given a genus $3$ canonical curve $X=\{ F=0\}$ we derive a set of equations for an open affine set of the Jacobian $J(X)$. The law group on the Jacobian is also explicitly constructed and, as an application, a set of equations for Kummer's variety $K(X)$ is obtained.

Explicit Kummer varieties of hyperelliptic Jacobian threefolds

AbstractWe explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a rational Weierstrass point defined over the same field. We also construct homogeneous quartic polynomials on the Kummer variety and show that they represent the duplication map using results of Stoll.Supplementary materials are available with this article.

On the equivariant algebraic Jacobian for curves of genus two

We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifies the structure of Flynn’s 72 defining quadratic relations. The treatment is also applied to the Kummer variety.

Characterizing Jacobians via trisecants of the Kummer variety

We prove Welters' trisecant conjecture: an indecomposable principally polarized abelian variety X is the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety K(X).

Combinatorial Properties of the K3 Surface: Simplicial Blowups and Slicings

The 4-dimensional abstract Kummer variety K 4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K 4, we resolve its 16 isolated singularities—step by step—by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL K3 surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover, we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the K3 surface of various topological types.

Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants

We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy.

PROJECTIVE EMBEDDINGS AND LAGRANGIAN FIBRATIONS OF KUMMER VARIETIES

It is known that holomorphic sections of an ample line bundle L (and its tensor power Lk) over an Abelian variety A are given by theta functions. Moreover, a natural basis of the space of holomorphic sections of L or Lk is related to a certain Lagrangian fibration of A. In our previous paper, we studied projective embeddings of A defined by these basis for Lk. For a natural torus action on the ambient projective space, it is proved that its moment map, restricted to A, approximates the Lagrangian fibration of A for large k, with respect to the "Gromov–Hausdorff topology". In this paper, we prove that the same is true for the Kummer variety associated to A.

A Generalization of the Dual Kummer Surface

Given a hyperelliptic curve \({\mathcal{C}}\) of genus g, we construct a variety \({\mathcal{k}}^*\) with a hyperplane section Hm whose complement \({\mathcal{k}}^*_m\) parametrizes classes of divisors on \({\mathcal{C}}\) of degree g + 1, in general position, modulo the ± Y involution. The variety \({\mathcal{k}}^*\) is birational to the Kummer variety belonging to \({\mathcal{C}}\). It is a tool for studying rational classes and divisors. The variety \({\mathcal{k}}^*_m\) is birational to a quotient of an affine variety by an affine group.

The moduli space of hyper-Kähler four-fold compactifications

I discuss some aspects of the moduli space of hyper-K{\"a}hler four-fold compactifications of type II and ${\cal M}$- theories. The dimension of the moduli space of these theories is strictly bounded from above. As an example I study Hilb$^2(K3)$ and the generalized Kummer variety $K^2(T^4)$. In both cases RR-flux (or $G$-flux in ${\cal M}$-theory) must be turned on, and we show that they give rise to vacua with ${\cal N}=2$ or ${\cal N}=3$ supersymmetry upon turning on appropriate fluxes. An interesting subtlety involving the symmetric product limit $S^2(K3)$ is pointed out.

Computing the Euler characteristic of generalized Kummer varieties

We give an elementary proof of the formula χ(KnA)=n3σ(n) for the Euler characteristic of the generalized Kummer variety KnA, where σ(n) denotes the sum of divisors function.

Lagrangian fibrations on generalized Kummer varieties

We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K n A is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K n A carries a divisor with vanishing Beauville-Bogomolov square.

Characterizing Jacobians via flexes of the Kummer Variety

Given an abelian variety $X$ and a point $a\in X$ we denote by $<a>$ the closure of the subgroup of $X$ generated by $a$. Let $N=2^g-1$. We denote by $\kappa: X\to \kappa(X)\subset\mathbb P^N$ the map from $X$ to its Kummer variety. We prove that an indecomposable abelian variety $X$ is the Jacobian of a curve if and only if there exists a point $a=2b\in X\setminus\{0\}$ such that $<a>$ is irreducible and $\kappa(b)$ is a flex of $\kappa(X)$.

On the elliptic genus of generalised Kummer varieties

Borisov and Libgober ([2]) recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde (see [6]) on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties [17] to deduce the following formula, conjectured by Kawai and Yoshioka ([15]), on the elliptic genus of a generalised Kummer variety A[[ n >]] of dimension 2(n−1): Here is the weak Jacobi form of weight −1 and index and V(n) is the Hecke operator sending Jacobi forms of index r to Jacobi forms of index nr (see [7]).

CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS

We discuss the conjecture of Buchstaber and Krichever that their multi-dimensional vector addition formula for Baker-Akhiezer functions characterizes Jacobians among principally polarized abelian varieties, and prove that it is indeed a weak characterization, i.e. that it is true up to additional components, or true precisely under a general position assumption. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety. We then use Buchstaber-Krichever's computation of the coefficients in the addition formula to obtain cubic relations among theta functions, which (weakly) characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are known classically by work of Mumford and Poor, but already for genus 4 they appear to be new.

The Coble hypersurfaces

Let A be an indecomposable principally polarized abelian variety of dimension g. Third order theta functions embed A in a projective space P(V 3) of dimension 3 g −1, while second order theta functions embed the Kummer variety X= A/{±1} in a projective space P(V 2) of dimension 2 g −1. Coble observed that for g=2 there is a unique cubic hypersurface in P(V 3) that is singular along A, and for g=3 a unique quartic hypersurface in P(V 2) singular along X. We explain these facts by a simple analysis of the representations of the corresponding Heisenberg group. To cite this article: A. Beauville, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Delzant models of moduli spaces

For every genus we construct a smooth, complete, rational polarized algebraic variety together with an effective normal crossing divisor such that for every moduli space of semistable topologically trivial vector bundles of rank 2 on an algebraic curve of genus there is a holomorphic isomorphism , where is the Kummer variety of the Jacobian of , sending the polarization of to the theta divisor of the moduli space. This isomorphism induces isomorphisms of the spaces and .

Kummer variety, geometry ofN-representability, and phase transitions

For a system of N indistinguishable fermions or bosons, we give a parametrization of the surface of a convex set of N-representable second-order density matrices, a polynomial equation for the exact energy levels, and an explanation of quantum phase transitions. We display the crucial role played by the Kummer variety as a graph of the 2-matrices for pure discrete states of an N-body system in quantum mechanics.