Géométrie torique computationelle
Org:
Charles Doran (Alberta),
Andrey Novoseltsev (Alberta) et
William Stein (Washington)
[
PDF]
 TRISTRAM BOGART, MSRI/San Francisco State University
Few smooth $d$polytopes with $N$ lattice points [PDF]

The AIM workshop "Combinatorial Challenges in Toric Varieties" centered on several questions about lattice polytopes motivated by toric geometry. In particular, a lattice polytope is said to be smooth if it is simple and the primitive rays of each vertex cone comprise a lattice basis. The questions are whether smoothness implies any or all of a list of successively stronger properties including normality, Koszul, and existence of a regular unimodular triangulation.
To gather evidence for these questions, we set out to test the properties systematically in dimension three. In order to do this, we proved a key finiteness property: given a finite list $F$ of rational cones and an integer $N$, there are only finitely many lattice polytopes whose normal cones are (integrally equivalent to) elements of $F$ and that contain at most $N$ lattice points. In particular this holds if $F$ simply consists of the unimodular cone, the smooth case. The proof is algorithmic and was implemented by Benjamin Lorenz to perform computations that give a positive answer to all of the questions for the case of threepolytopes with at most twelve lattice points. In algebrogeometric language, the strongest result is that if $X$ is a toric threefold embedded in $\mathbb{P}^{N1}$ by a complete linear series for $N \leq 12$, then the defining ideal $I(X)$ has a squarefree quadratic initial ideal.
This project is joint work with Christian Haase, Milena Hering, Lorenz, Benjamin Nill, Andreas Paffenholz, Francisco Santos, and Hal Schenck.
 VINCENT BOUCHARD, University of Alberta
Tops [PDF]

It is well known that reflexive polytopes can be used to study CalabiYau hypersurfaces in toric varieties. Less however is known about the use of "tops"  threedimensional lattice polytopes with one facet containing the origin and the other facets at integral distance one from the origin. Those appear in the context of elliptically fibered CalabiYau hypersurfaces, where they encode degenerations of elliptic fibers. In this talk I will summarize the uses of tops, and explain how they can be classified into families closely related to the classification of affine KacMoody algebras.
 VOLKER BRAUN, Dublin Institute for Advanced Studies
Toric Geometry in Sage [PDF]

In this talk, I will present the toric geometry package that I
developed jointly with Andrey Novoseltsev. It is implemented in the
Sage computer mathematics system and provides an efficient and
userfriendly implementation of many toric algorithms. As an
application, I will construct an elliptically fibered fourfold whose
discriminant locus contains a nonsimply connected divisor. This is
relevant for Ftheory compactifications of string theory.
 CHARLES DORAN, Alberta
 ANDREY NOVOSELTSEV, University of Alberta
Geometric transitions through singular subfamilies in toric varieties [PDF]

(Joined work with Charles F. Doran.) We present an explicit geometric transition between
CalabiYau threefolds realized as anticanonical hypersurfaces and
complete intersections in toric varieties, where the role of the singular
variety in the geometric transition is played by a generically
singular subfamily of hypersurfaces. The involved families of
CalabiYau varieties may lead to a representative for the last class
in DoranMorgan classification of variations of Hodge structure
associated to oneparameter families of CalabiYau threefolds. In the
exploration of this example we have extensively used the newly
developed framework for toric geometry in Sage.
 URSULA WHITCHER, Harvey Mudd College
The GriffithsDwork algorithm for toric hypersurfaces [PDF]

The variation of complex structure for a family of CalabiYau varieties is encoded in a differential equation called the PicardFuchs equation. We develop computational methods for studying properties of CalabiYau varieties realized as hypersurfaces in toric varieties, and implement an algorithm for computing their PicardFuchs equations. We apply our methods to study highly symmetric families of K3 surfaces. This talk describes joint work with Dagan Karp, Jacob Lewis, Daniel Moore, and Dmitri Skjorshammer.
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