28 research outputs found
Tits buildings and K-stability
A polarized variety is K-stable if, for any test configuration, the
Donaldson-Futaki invariant is positive. In this paper, inspired by classical
geometric invariant theory, we describe the space of test configurations as a
limit of a direct system of Tits buildings. We show that the Donaldson-Futaki
invariant, conveniently normalized, is a continuous function on this space. We
also introduce a pseudo-metric on the space of test configurations. Recall that
K-stability can be enhanced by requiring that the Donaldson-Futaki invariant is
positive on any admissible filtration of the co-ordinate ring. We show that
admissible filtrations give rise to Cauchy sequences of test configurations
with respect to the above mentioned pseudo-metric.Comment: 16 pages. To appear on the Proceedings of the Edinburgh Mathematical
Societ
Hyperelliptic Schottky Problem and Stable Modular Forms
It is well known that, fixed an even, unimodular, positive definite quadratic
form, one can construct a modular form in each genus; this form is called the
theta series associated to the quadratic form. Varying the quadratic form, one
obtains the ring of stable modular forms. We show that the differences of theta
series associated to specific pairs of quadratic forms vanish on the locus of
hyperelliptic Jacobians in each genus. In our examples, the quadratic forms
have rank 24, 32 and 48. The proof relies on a geometric result about the
boundary of the Satake compactification of the hyperelliptic locus. We also
study the monoid formed by the moduli space of all principally polarised
abelian varieties, the operation being the product of abelian varieties. We use
this construction to show that the ideal of stable modular forms vanishing on
the hyperelliptic locus in each genus is generated by differences of theta
series.Comment: Final version, title changed, published in Documenta Mathematic
Moduli and Periods of Supersymmetric Curves
Supersymmetric curves are the analogue of Riemann surfaces in super geometry.
We establish some foundational results about complex Deligne-Mumford
superstacks, and we then prove that the moduli superstack of supersymmetric
curves is a smooth complex Deligne-Mumford superstack. We then show that the
superstack of supersymmetric curves admits a coarse complex superspace, which,
in this case, is just an ordinary complex space. In the second part of this
paper we discuss the period map. We remark that the period domain is the moduli
space of ordinary abelian varieties endowed with a symmetric theta divisor, and
we then show that the differential of the period map is surjective. In other
words, we prove that any first order deformation of a classical Jacobian is the
Jacobian of a supersymmetric curve.Comment: Minor revision, to appear on Advances in Theoretical and Mathematical
Physic
The degree of the Gauss map of the theta divisor
We study the degree of the Gauss map of the theta divisor of principally
polarised complex abelian varieties. We use this to obtain a bound on the
multiplicity of the theta divisor along irreducible components of its singular
locus, and apply this bound in examples, and to understand the local structure
of isolated singular points. We further define a stratification of the moduli
space of ppav's by the degree of the Gauss map. In dimension four, we show that
this stratification gives a weak solution of the Schottky problem, and we
conjecture that this is true in any dimension
On some modular contractions of the moduli space of stable pointed curves
The aim of this paper is to study some modular contractions of the moduli
space of stable pointed curves. These new moduli spaces, which are modular
compactifications of the moduli space of smooth pointed curves, are related
with the minimal model program for the moduli space of stable pointed curves
and have been introduced in a previous work of the authors. We interpret them
as log canonical models of adjoints divisors and we then describe the Shokurov
decomposition of a region of boundary divisors on the moduli space of stable
pointed curves.Comment: 30 pages, 1 figure. To appear on Algebra and Number Theor
The Gauss map and secants of the Kummer variety
Fay's trisecant formula shows that the Kummer variety of the Jacobian of a
smooth projective curve has a four dimensional family of trisecant lines. We
study when these lines intersect the theta divisor of the Jacobian, and prove
that the Gauss map of the theta divisor is constant on these points of
intersection, when defined. We investigate the relation between the Gauss map
and multisecant planes of the Kummer variety as well.Comment: Minor changes, to appear on the Bulletin of London Mathematical
Societ
Semicontinuity of Gauss maps and the Schottky problem
We show that the degree of Gauss maps on abelian varieties is semicontinuous
in families, and we study its jump loci. As an application we obtain that in
the case of theta divisors this degree answers the Schottky problem. Our proof
computes the degree of Gauss maps by specialization of Lagrangian cycles on the
cotangent bundle. We also get similar results for the intersection cohomology
of varieties with a finite morphism to an abelian variety; it follows that many
components of Andreotti-Mayer loci, including the Schottky locus, are part of
the stratification of the moduli space of ppav's defined by the topological
type of the theta divisor.Comment: Final version, to appear in Math. Annale
A note on families of K-semistable log-Fano pairs
In this short note, we give an alternative proof of the semipositivity of the
Chow-Mumford line bundle for families of K-semistable log-Fano pairs, and of
the nefness threeshold for the log-anti-canonical line bundle on families of
K-stable log Fano pairs. We also prove a bound on the multiplicity of fibers
for families of K-semistable log Fano varieties, which to the best of our
knowledge is new.Comment: comments are welcome
Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach
We give completely algebro-geometric proofs of a theorem by T. Shiota, and of
a theorem by I. Krichever, characterizing Jacobians of algebraic curves among
all irreducible principally polarized abelian varieties. Shiota's
characterization is given in terms of the KP equation. Krichever's
characterization is given in terms of trisecant lines to the Kummer variety.
Here we treat the case of flexes and degenerate trisecants. The basic tool we
use is a theorem we prove asserting that the base locus of the linear system
associated to an effective line bundle on an abelian variety is reduced. This
result allows us to remove all the extra assumptions that were introduced in
the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in
order to achieve algebro-geometric proofs of the results above.Comment: 21 page