34 research outputs found
Torelli problem for Calabi-Yau threefolds with GLSM description
We construct a gauged linear sigma model with two non-birational K\"alher
phases which we prove to be derived equivalent, -equivalent,
deformation equivalent and Hodge equivalent. This provides a new counterexample
to the birational Torelli problem which admits a simple GLSM interpretation.Comment: Exposition improved, some arguments clarifie
Special lines on contact manifolds
In a series of two articles Kebekus studied deformation theory of
minimal rational curves on contact Fano manifolds. Such curves are called contact
lines. Kebekus proved that a contact line through a general point is necessarily
smooth and has a fixed standard splitting type of the restricted tangent bundle. In
this paper we study singular contact lines and those with special splitting type. We
provide restrictions on the families of such lines, and on contact Fano manifolds
which have reducible varieties of minimal rational tangents. We also show that
the results about singular lines naturally generalise to complex contact manifolds,
which are not necessarily Fano, for instance, quasi-projective contact manifolds
or compact contact manifolds of Fujiki class C. In particular, in many cases the
dimension of a family of singular lines is at most 2 less than the dimension of the
contact manifold
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
Symmetric locally free resolutions and rationality problems
We study symmetric locally free resolutions of a two torsion sheaf, or theta
characteristic, on a plane curve. More specifically, we show that two different
locally free resolutions of a sheaf give rise to two quadric bundles that are
birational to one another, as long as their ranks are the same. We construct
explicit models of such quadric bundles with given discriminant data.
In particular we deduce various unexpected birational models of a nodal
Gushel-Mukai fourfold, as well as of a cubic fourfold containing a plane. As an
application, we discuss stable-rationality of very general quadric bundles over
with discriminant curve of fixed degree. We deduce also the non
stable-rationality of several complete intersections of small degree.Comment: 27 pages, improved versio
Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections
We study Kustin-Miller unprojections between Calabi-Yau threefolds or more
precisely the geometric transitions they induce. We use them to connect many
families of Calabi-Yau threefolds with Picard number one to the web of Calabi
Yau complete intersections. This enables us to find explicit description of a
few known families of Calabi-Yau threefolds in terms of equations. Moreover we
find two new examples of Calabi-Yau threefolds with Picard group of rank one,
described by Pfaffian equations in weighted projective spaces.Comment: to appear in Journal of Geometry and Physic
Special lines on contact manifolds
In a series of two articles Kebekus studied deformation theory of
minimal rational curves on contact Fano manifolds. Such curves are called contact
lines. Kebekus proved that a contact line through a general point is necessarily
smooth and has a fixed standard splitting type of the restricted tangent bundle. In
this paper we study singular contact lines and those with special splitting type. We
provide restrictions on the families of such lines, and on contact Fano manifolds
which have reducible varieties of minimal rational tangents. We also show that
the results about singular lines naturally generalise to complex contact manifolds,
which are not necessarily Fano, for instance, quasi-projective contact manifolds
or compact contact manifolds of Fujiki class C. In particular, in many cases the
dimension of a family of singular lines is at most 2 less than the dimension of the
contact manifold