559 research outputs found
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
Let X -> Y be a fibration whose fibers are complete intersections of two
quadrics. We develop new categorical and algebraic tools---a theory of relative
homological projective duality and the Morita invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting---to study
semiorthogonal decompositions of the bounded derived category of X. Together
with new results in the theory of quadratic forms, we apply these tools in the
case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers
are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds,
respectively. In the latter two cases, if Y is the projective line over an
algebraically closed field of characteristic zero, we relate rationality
questions to categorical representability of X.Comment: 43 pages, changes made and some material added and corrected in
sections 1, 4, and 5; this is the final version accepted for publication at
Journal de Math\'ematiques Pures et Appliqu\'ee
Primitive contractions of Calabi-Yau threefolds II
We construct 16 new examples of Calabi--Yau threefolds with Picard group of
rank 1. Each of these examples is obtained by smoothing the image of a
primitive contraction with exceptional divisor being a del Pezzo surface of
degree 6, 7 or .Comment: 20 pages, to appear in JLM
Projective geometry of Wachspress coordinates
We show that there is a unique hypersurface of minimal degree passing through
the non-faces of a polytope which is defined by a simple hyperplane
arrangement. This generalizes the construction of the adjoint curve of a
polygon by Wachspress in 1975. The defining polynomial of our adjoint
hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a
key ingredient for the definition of Wachspress coordinates, which are
barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial
also appears both in algebraic statistics, when studying the moments of uniform
probability distributions on polytopes, and in intersection theory, when
computing Segre classes of monomial schemes. We describe the Wachspress map,
the rational map defined by the Wachspress coordinates, and the Wachspress
variety, the image of this map. The inverse of the Wachspress map is the
projection from the linear span of the image of the adjoint hypersurface. To
relate adjoints of polytopes to classical adjoints of divisors in algebraic
geometry, we study irreducible hypersurfaces that have the same degree and
multiplicity along the non-faces of a polytope as its defining hyperplane
arrangement. We list all finitely many combinatorial types of polytopes in
dimensions two and three for which such irreducible hypersurfaces exist. In the
case of polygons, the general such curves< are elliptic. In the
three-dimensional case, the general such surfaces are either K3 or elliptic
Complete intersections: Moduli, Torelli, and good reduction
We study the arithmetic of complete intersections in projective space over
number fields. Our main results include arithmetic Torelli theorems and
versions of the Shafarevich conjecture, as proved for curves and abelian
varieties by Faltings. For example, we prove an analogue of the Shafarevich
conjecture for cubic and quartic threefolds and intersections of two quadrics.Comment: 37 pages. Typo's fixed. Expanded Section 2.
Projections of del Pezzo surfaces and Calabi--Yau threefolds
We study the syzygetic structure of projections of del Pezzo surfaces in
order to construct singular Calabi-Yau threefolds. By smoothing those
threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group
of rank 1. We also give an example of type II primitive contraction whose
exceptional divisor is the blow-up of the projective plane at a point.Comment: A table of known Calabi--Yau threefolds with Picard number 1 is
added, to appear in Advances in Geometr
Examples of cylindrical Fano fourfolds
We construct 4 di erent families of smooth Fano fourfolds with Picard rank 1,
which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z
is a quasiprojective variety. The affi ne cones over such a fourfold admit eff
ective Ga-actions. Similar constructions of cylindrical Fano threefolds were
done previously in our papers jointly with Takashi Kishimoto.Comment: 23
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