890 research outputs found
A very special EPW sextic and two IHS fourfolds
We show that the Hilbert scheme of two points on the Vinberg surface has
a 2:1 map onto a very symmetric EPW sextic in . The fourfold
is singular along planes, of which form a complete family of
incident planes. This solves a problem of Morin and O'Grady and establishes
that is the maximal cardinality of such a family of planes. Next, we show
that this Hilbert scheme is birationally isomorphic to the Kummer type IHS
fourfold constructed in [DW]. We find that is also related to the
Debarre-Varley abelian fourfold.Comment: 32 page
Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary
Let be the -punctured disk. We prove that a family of essential
simple arcs starting and ending at the boundary and pairwise intersecting at
most twice is of size at most . On the way, we also show that
any nontrivial square complex homeomorphic to a disk whose hyperplanes are
simple arcs intersecting at most twice must have a corner or a spur.Comment: Assaf Bar-Natan's MSc thesis, written under the supervision of Prof.
Piotr Przytyck
Compactifications defined by arrangements II: locally symmetric varieties of type IV
We define a new class of completions of locally symmetric varieties of type
IV which interpolates between the Baily-Borel compactification and Mumford's
toric compactifications. An arithmetic arrangement in a locally symmetric
variety of type IV determines such a completion canonically. This completion
admits a natural contraction that leaves the complement of the arrangement
untouched. The resulting completion of the arrangement complement is very much
like a Baily-Borel compactification: it is the proj of an algebra of
meromorphic automorphic forms. When that complement has a moduli space
interpretation, then what we get is often a compactification obtained by means
of geometric invariant theory. We illustrate this with several examples: moduli
spaces of polarized and Enriques surfaces and the semi-universal
deformation of a triangle singularity.
We also discuss the question when a type IV arrangement is definable by an
automorphic form.Comment: The section on arrangements on tube domains has beeen expanded in
order to make a connection with a conjecture of Gritsenko and Nikulin. Also
added: a list of notation and some references. Finally some typo's corrected
and a few minor changes made in notatio
Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms
The thirty years old programme of Griffiths and Harris of understanding
higher-dimensional analogues of Poncelet-type problems and synthetic approach
to higher genera addition theorems has been settled and completed in this
paper. Starting with the observation of the billiard nature of some classical
constructions and configurations, we construct the billiard algebra, that is a
group structure on the set T of lines in simultaneously tangent to d-1
quadrics from a given confocal family. Using this tool, the related results of
Reid, Donagi and Knoerrer are further developed, realized and simplified. We
derive a fundamental property of T: any two lines from this set can be obtained
from each other by at most d-1 billiard reflections at some quadrics from the
confocal family. We introduce two hierarchies of notions: s-skew lines in T and
s-weak Poncelet trajectories, s = -1,0,...,d-2. The interrelations between
billiard dynamics, linear subspaces of intersections of quadrics and
hyperelliptic Jacobians developed in this paper enabled us to obtain
higher-dimensional and higher-genera generalizations of several classical genus
1 results: the Cayley's theorem, the Weyr's theorem, the Griffiths-Harris
theorem and the Darboux theorem.Comment: 36 pages, 11 figures; to be published in Advances in Mathematic
Darboux cyclides and webs from circles
Motivated by potential applications in architecture, we study Darboux
cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin
cyclides and quadrics, and they carry up to six real families of circles.
Revisiting the classical approach to these surfaces based on the spherical
model of 3D Moebius geometry, we provide computational tools for the
identification of circle families on a given cyclide and for the direct design
of those. In particular, we show that certain triples of circle families may be
arranged as so-called hexagonal webs, and we provide a complete classification
of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure
Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian
We prove the existence of various families of irreducible homaloidal
hypersurfaces in projective space , for all . Some of
these are families of homaloidal hypersurfaces whose degrees are arbitrarily
large as compared to the dimension of the ambient projective space. The
existence of such a family solves a question that has naturally arisen from the
consideration of the classes of homaloidal hypersurfaces known so far. The
result relies on a fine analysis of dual hypersurfaces to certain scroll
surfaces. We also introduce an infinite family of determinantal homaloidal
hypersurfaces based on a certain degeneration of a generic Hankel matrix. These
examples fit non--classical versions of de Jonqui\`eres transformations. As a
natural counterpoint, we broaden up aspects of the theory of Gordan--Noether
hypersurfaces with vanishing Hessian determinant, bringing over some more
precision to the present knowledge.Comment: 56 pages. Some material added in section 1; minor changes. Final
version to appear in Advances in Mathematic
The Brauer-Manin Obstruction and Sha[2].
We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of Sha[2], for families of curves of genus 2 of the form y^2 = f(x), where f(x) is a quintic containing an irreducible cubic factor
- …