Related papers: Picard-Fuchs equations and mirror maps for hypersu…
We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…
We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the…
This paper intends to give a mathematical explanation for results on the zeta-function of some families of varieties recently obtained in the context of Mirror Symmetry. In doing so, we obtain concrete and explicit examples for some results…
In this paper, we give a formula for the number of rational points on the Dwork hypersurfaces of degree six over finite fields by using Greene's finite-field hypergeometric function, which is a generalization of Goodson's formula for the…
We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge…
Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…
The general quintic hypersurface in ${\mathbb P}^4$ is the most famous example of a Calabi--Yau threefold for which mirror symmetry has been investigated in detail. There is a description of the mirror as a hypersurface in a certain…
We use Maxima to compute the moment matrices of hypersurfaces. After that, we compute the Hilbert-Mumford numerical criterion for plane cubics and plane quartics, and give the stability of these curves.
Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified…
A new method to obtain the Picard-Fuchs equations of effective $N = 2$ supersymmetric gauge theories in 4 dimensions is developed. It includes both pure super Yang-Mills and supersymmetric gauge theories with massless matter…
Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces…
Let $k$ be either a number a field or a function field over $\mathbb{Q}$ with finitely many variables. We present a practical algorithm to compute the geometric Picard lattice of a K3 surface over $k$ of degree $2$, i.e., a double cover of…
We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard--Fuchs differential equations; we count points using Gauss sums and…
In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those…
We show how certain F^4 couplings in eight dimensions can be computed using the mirror map and K3 data. They perfectly match with the corresponding heterotic one-loop couplings, and therefore this amounts to a successful test of the…
In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…
Geometry of hypersurfaces defined by the relation which generalizes classical formula for free energy in terms of microstates is studied. Induced metric, Riemann curvature tensor, Gauss-Kronecker curvature and associated entropy are…
In this work, we give a purely analytic introduction to the phenomenon of mirror symmetry for quintic threefolds via classical hypergeometric functions and differential equations for them. Starting with a modular map and recent…
We compute Yukawa couplings involving chiral matter fields in toroidal compactifications of higher dimensional super-Yang-Mills theory with magnetic fluxes. Specifically we focus on toroidal compactifications of D=10 super-Yang-Mills…
We argue that global F-theory compactifications to four dimensions generally exhibit higher rank Yukawa matrices from multiple geometric contributions known as Yukawa points. The holomorphic couplings furthermore have large hierarchies for…