1,591 research outputs found
Computing Periods of Hypersurfaces
We give an algorithm to compute the periods of smooth projective
hypersurfaces of any dimension. This is an improvement over existing algorithms
which could only compute the periods of plane curves. Our algorithm reduces the
evaluation of period integrals to an initial value problem for ordinary
differential equations of Picard-Fuchs type. In this way, the periods can be
computed to extreme-precision in order to study their arithmetic properties.
The initial conditions are obtained by an exact determination of the cohomology
pairing on Fermat hypersurfaces with respect to a natural basis.Comment: 33 pages; Final version. Fixed typos, minor expository changes.
Changed code repository lin
Threefold Flops via Matrix Factorization
The explicit McKay correspondence, as formulated by Gonzalez-Sprinberg and
Verdier, associates to each exceptional divisor in the minimal resolution of a
rational double point a matrix factorization of the equation of the rational
double point. We study deformations of these matrix factorizations, and show
that they exist over an appropriate "partially resolved" deformation space for
rational double points of types A and D. As a consequence, all simple flops of
lengths 1 and 2 can be described in terms of blowups defined from matrix
factorizations. We also formulate conjectures which would extend these results
to rational double points of type E and simple flops of length greater than 2.Comment: v2: minor change
Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries
At special loci in their moduli spaces, Calabi-Yau manifolds are endowed with
discrete symmetries. Over the years, such spaces have been intensely studied
and have found a variety of important applications. As string compactifications
they are phenomenologically favored, and considerably simplify many important
calculations. Mathematically, they provided the framework for the first
construction of mirror manifolds, and the resulting rational curve counts.
Thus, it is of significant interest to investigate such manifolds further. In
this paper, we consider several unexplored loci within familiar families of
Calabi-Yau hypersurfaces that have large but unexpected discrete symmetry
groups. By deriving, correcting, and generalizing a technique similar to that
of Candelas, de la Ossa and Rodriguez-Villegas, we find a calculationally
tractable means of finding the Picard-Fuchs equations satisfied by the periods
of all 3-forms in these families. To provide a modest point of comparison, we
then briefly investigate the relation between the size of the symmetry group
along these loci and the number of nonzero Yukawa couplings. We include an
introductory exposition of the mathematics involved, intended to be accessible
to physicists, in order to make the discussion self-contained.Comment: 54 pages, 3 figure
K3-fibered Calabi-Yau threefolds I, the twist map
A construction of Calabi-Yaus as quotients of products of lower-dimensional
spaces in the context of weighted hypersurfaces is discussed, including
desingularisation. The construction leads to Calabi-Yaus which have a fiber
structure, in particular one case has K3 surfaces as fibers. These Calabi-Yaus
are of some interest in connection with Type II -heterotic string dualities in
dimension 4. A section at the end of the paper summarises this for the
non-expert mathematician.Comment: 31 pages LaTeX, 11pt, 2 figures. To appear in International Journal
of Mathematics. On the web at
http://personal-homepages.mis.mpg.de/bhunt/preprints.html , #
A note on graded hypersurface singularities
For a weighted quasihomogeneous two dimensional hypersurface singularity, we
define a smoothing with unipotent monodromy and an isolated graded normal
singularity. We study the natural weighted blow up of both the smoothing and
the surface. In particular, we describe our construction for the
quasihomogeneous singularities of type I, the 14 unimodal exceptional
singularities and we relate it to their stable replacement.Comment: 19 pages, 4 figures. New section on stable replacement of Unimodal
singularities, Several improvements and corrections. Comments welcom
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