3,433 research outputs found
Complete Intersection Fibers in F-Theory
Global F-theory compactifications whose fibers are realized as complete
intersections form a richer set of models than just hypersurfaces. The detailed
study of the physics associated with such geometries depends crucially on being
able to put the elliptic fiber into Weierstrass form. While such a
transformation is always guaranteed to exist, its explicit form is only known
in a few special cases. We present a general algorithm for computing the
Weierstrass form of elliptic curves defined as complete intersections of
different codimensions and use it to solve all cases of complete intersections
of two equations in an ambient toric variety. Using this result, we determine
the toric Mordell-Weil groups of all 3134 nef partitions obtained from the 4319
three-dimensional reflexive polytopes and find new groups that do not exist for
toric hypersurfaces. As an application, we construct several models that cannot
be realized as toric hypersurfaces, such as the first toric SU(5) GUT model in
the literature with distinctly charged 10 representations and an F-theory model
with discrete gauge group Z_4 whose dual fiber has a Mordell-Weil group with
Z_4 torsion.Comment: 41 pages, 4 figures and 18 tables; added references in v
A Las Vegas algorithm to solve the elliptic curve discrete logarithm problem
In this paper, we describe a new Las Vegas algorithm to solve the elliptic
curve discrete logarithm problem. The algorithm depends on a property of the
group of rational points of an elliptic curve and is thus not a generic
algorithm. The algorithm that we describe has some similarities with the most
powerful index-calculus algorithm for the discrete logarithm problem over a
finite field
Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
We give examples over arbitrary fields of rings of invariants that are not
finitely generated. The group involved can be as small as three copies of the
additive group, as in Mukai's examples over the complex numbers. The failure of
finite generation comes from certain elliptic fibrations or abelian surface
fibrations having positive Mordell-Weil rank.
Our work suggests a generalization of the Morrison-Kawamata cone conjecture
from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in
dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic
Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
Recently, Cao-Maulik-Toda defined stable pair invariants of a compact
Calabi-Yau 4-fold . Their invariants are conjecturally related to the
Gopakumar-Vafa type invariants of defined using Gromov-Witten theory by
Klemm-Pandharipande. In this paper, we consider curve counting invariants of
using Hilbert schemes of curves and conjecture a DT/PT correspondence which
relates these to stable pair invariants of .
After providing evidence in the compact case, we define analogous invariants
for toric Calabi-Yau 4-folds using a localization formula. We formulate a
vertex formalism for both theories and conjecture a relation between the (fully
equivariant) DT/PT vertex, which we check in several cases. This relation
implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary
insertions.Comment: 28 pages. Published versio
Discriminants in the Grothendieck Ring
We consider the "limiting behavior" of *discriminants*, by which we mean
informally the locus in some parameter space of some type of object where the
objects have certain singularities. We focus on the space of partially labeled
points on a variety X, and linear systems on X. These are connected --- we use
the first to understand the second. We describe their classes in the
Grothendieck ring of varieties, as the number of points gets large, or as the
line bundle gets very positive. They stabilize in an appropriate sense, and
their stabilization is given in terms of motivic zeta values. Motivated by our
results, we conjecture that the symmetric powers of geometrically irreducible
varieties stabilize in the Grothendieck ring (in an appropriate sense). Our
results extend parallel results in both arithmetic and topology. We give a
number of reasons for considering these questions, and propose a number of new
conjectures, both arithmetic and topological.Comment: 39 pages, updated with progress by others on various conjecture
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