Special Year on Algebraic Geometry and Topology
with partial financial support provided by the
Australian Mathematical Sciences
Institute
Abstracts for Minimal Models Activity
Monday 18 August - Friday 29 August, 2003
Click on the title of a talk to find its position in the
program.
Ambro: Parabolic log varieties
I will discuss positivity properties of families of log minimal models of
Kodaira dimension zero, with applications to the structure of such varieties.
Buckley: Orbifold RR for polarised threefolds
An orbifold RR formula for polarised 3-folds with prescribed singular
locus (An curves and quotient points) is stated and proved. The singular
locus induces explicit contributions to the RR-type expression. The formula
can be used for a systematic computer search of CY threefolds. A few
interesting new examples of CY 3-folds are found.
de Fernex: Bounds on log canonical thresholds, with applications to
birational rigidity
We begin by presenting an inequality that relates log canonical threshold
and Samuel multiplicity of zero dimensional ideals, and by discussing how,
under certain assumptions, the log canonical threshold of a pair can be
controlled by "projecting" the pair through a suitable smooth morphism. The
bounds on log canonical thresholds that are obtained by these means are
then applied to prove the birational rigidity of smooth hypersurfaces of
degree N in PN, for
4 < N < 12. The results presented in this talk are
joint works with L. Ein and M. Mustata.
de Jong: Stable orders over surfaces
This is report on joint work with Michael Artin. The talk will discuss the
birational classification of "noncommutative surfaces finite over their
center" that has been worked out by Artin and others in the field of
noncommutative algebraic geometry. It turns out to be closely related to
the birational geometry of log surfaces (even though I won't discuss this
simply because I don't know enough). On the other hand, somehow the
geometry of the order over the surface restricts the possibilities that
can occur thereby producing a nice classification.
de Jong: Gabber's Theorem Br X = Br' X
for quasiprojective schemes X
Some time ago Gabber proved the result above which compares the
cohomological Brauer group
Br' X = H2(X,
Gm)tor to the actual Brauer
group Br X (defined using Azumaya algebras). He never published the proof. I
recently discovered another proof of this result (which appears to be much
easier). It uses the yoga of so-called "alpha-twisted" sheaves, which
simplifies many arguments related to this question.
de Jong: Period = Index for Brauer classes on surfaces
An element in the Brauer group of the function field of a surface over an
algebraically closed field can be represented by a division algebra of
degree (=index) equal to the order (=period) of the element in the Brauer
group. My proof requires the condition that the order should be prime to
the characteristic of the ground field. In fact one can prove the following
result: For a nonsingular surface U over an algebraically closed field, and
n prime to the characteristic, the boundary map in étale cohomology
H1(U, PGLn) ->
H2(U,µn)
coming from the exact sequence of group schemes
1 ->µn -> SLn -> PGLn
-> 1
is surjective. The proof uses geometric arguments.
For Brauer groups of function fields the following questions are open:
Is every division algebra over the function field of a surface cyclic?
I expect the answer to be: No.
Over a function field of transcendence degree d:
Does every element of
period n have index dividing nd-1? This seems plausible, allthough I
do not have even a heuristic argument.
Dolgachev: A complex ball uniformization of the moduli space of cubic
surfaces via periods of K3 surfaces
Getzler:
Nilpotent Lie algebras and relations in the cohomology of
Deligne-Mumford moduli spaces
The cohomology of certain nilpotent Lie algebras associated to symplectic
vector spaces may be calculated using (at least partially) Kostant's
Laplacian. This cohomology occurs in analysis by means of Hodge theory of a
spectral sequence for the cohomology of the Deligne-Mumford-Knudsen moduli
space of stable n-pointed curves of genus g. In this talk, I work out in
detail the case g=1, and explain why higher genus is more complicated, and
what we can and cannot hope to generalise.
Hassett: Towards a canonical model for the moduli space of
curves
This is joint with D. Hyeon. Consider the moduli space of stable curves as
a log-variety, with boundary delta corresponding to the nodal curves. We
seek to describe its log canonical model with respect to
K + A delta.
When A=1, we recover the moduli space of stable curves; for A=0, this would
be the canonical model of the moduli space, which is expected to exist for
g>23 by work of Eisenbud, Harris, and Mumford. For intermediate values of
A, the log canonical model can be constructed with Geometric Invariant
Theory. As A decreases, the log canonical model parametrizes curves with
increasingly complicated singularities: cusps, tacnodes, and worse.
Hayakawa: Divisors with minimal discrepancy over 3-dimensional terminal
singularities
We study blowing ups of 3-dimensional terminal singularities such that their
exceptional divisors are prime and have minimal discrepancies. Various
examples will be provided.
Katsura: Invariants of algebraic varieties
in positive characteristic
We introduce two invariants of algebraic varieties in positive
characteristic. One is a generalization of the a-number of an abelian
variety, and the other is a generalization of the height of the Artin-Mazur
formal group of a Calabi-Yau variety. We make clear the meanings of these
invariants and calculate them in some examples. We also explain that the
Artin conjecture for K3 surfaces does not hold for higher dimensional
Calabi-Yau varieties.
Kawakita: Divisorial contractions
The minimal model program has been formulated to generalise
the theory of minimal models of surfaces to higher dimensional varieties.
For a given variety, it produces a good variety after a finite sequence of
elementary transformations called divisorial contractions and flips.
Since Mori completed this program in dimension three by proving the existence
of three-fold flips, we have hoped the explicit study of three-folds
to strengthen our grasp of the theory of minimal models in higher dimension
as well as three-folds themselves.
We complete the explicit study of a three-fold divisorial contraction
whose exceptional divisor contracts to a point.
Kudryavtsev: Three-dimensional contractions and log del Pezzo surfaces
We will discuss the inductive method of algebraic varieties classification
and prove the easy part of the "weak general elephant conjecture".
Neeman: A non-commutative generalisation of the problem of extending vector
bundles
It has been known for a long time that coherent sheaves can be extended from
an open subset of a scheme, but vector bundles cannot. Not even if by
"extensions" we understand extending to a chain complex of vector bundles,
up to quasi-isomorphism. An important theorem of Thomason's says that any
vector bundle is a direct summand of a complex of vector bundles which does
extend. This allows one to get a long exact sequence in the algebraic
K-theory of possibly singular schemes. We will explain this.
It turns out to be interesting to look at the non-commutative analogue. It
comes up naturally in topological applications. We will explain the problem,
and the recent work by Ranicki and the speaker.
Pasarescu: Curves on rational surfaces
with hyperelliptic hyperplane sections
A classical problem from the theory of projective curves consist in the
determination, for any given integer n>2, of all possible pairs
(d,g) for
which there is a smooth, irreducible, non-degenerate curve of degree d and
genus g in Pn. We solve the problem for the
non-lacunary domain (based on
some previous results of Ciliberto, Sernesi and the author also) by
constructing curves on rational surfaces with hyperelliptic hyperplane
sections. We give also a Conjecture for the lacunary domain which deals with
curves on scrolls also.
Pedrini: Finite dimensional motives and
Bloch-Beilinson's conjectures
We show how the finite dimensionality of the motive M(X) of a smooth
variety X (in the triangulated category defined by Voevodsky) is related
to the existence of a filtration on the Chow groups of X. For surfaces
with geometric genus 0 finite dimensionality is equivalent to Bloch's
conjecture on the vanishing of the Albanese kernel.
Reid: Unprojection
The ideas of projection and unprojection play a prominent role in the
biregular and birational geometry of Fano 3-folds. I discuss the
Kustin-Miller unprojection theorem in the interpretation of
Papadakis-Reid (see math.AG/0011094) together with its applications.
There is a conjectural generalisation to the case the unprojected
subscheme D in X is only assumed quasi-Gorenstein.
See also my
"Graded rings and birational geometry", my
website
+ 3-fold links.
Sebestean: McKay correspondence,
examples and calculations
I will discuss some basic ideas on the McKay correspondence, starting with
examples in dimension 2 and relation with the ADE diagrams. The 3-dimensional
case, especially A-Hilb for A in SL(3,C) an Abelian subgroup
according to
Craw and Reid. The general result of [BKR] in terms of derived categories,
and some examples and counterexamples in dimension >4.
Starr: Sections of rationally connected fibrations
Every rationally connected fibration over a curve (over an algebraically
closed field) has a section. I will discuss the proof of this theorem, some
consequences, and a converse to this theorem. This is joint work with
A.J. de Jong, T. Graber, J. Harris and B. Mazur.
Starr: Rational curves on Fano hypersurfaces
This talk will describe joint research with A. J. de Jong, Joe Harris and
Mike Roth on the geometry of the spaces parametrizing rational curves on
Fano hypersurfaces, in particular the irreducibility of these spaces and
their Kodaira dimensions. As will be explained, this research is
motivated by two open problems:
giving examples of smooth Fano hypersurfaces which are not unirational,
and
generalizing Lang's theorem that a hypersurface of degree d in
Pn
defined over the function field of a surface has a rational point if
d2 < n.
Suzuki: On Fano indices of Q-Fano 3-folds
We prove that the Fano index of a Q-Fano 3-fold is contained
in the set
{1, 2, ..., 9, 10, 11, 13, 17, 19}. With exception of
f = 10, all the values
occur. For example, PP(3,4,5,7) is a Q-Fano 3-fold with
f = 19. We suspect
that the case f = 10 does not occur. The methods
we use include Kawamata's
boundedness argument, the singular orbifold theorem of Reid, and Magma
computer programs kindly provided by Gavin Brown.
van Hamel: A homology reinterpretation
of the Brauer-Manin obstruction in the local-global problem for
varieties over function fields
As was observed by Manin, the Brauer group of a variety over a global field
provides an obstruction to approximate local points by a global rational
point. It is hoped that for rationally connected varieties this is the only
obstruction. I will explain the Brauer-Manin obstruction and present a
reinterpretation in terms of a "pseudo-motivic" homology theory. Then I will
try to convey my belief that this reinterpretation might be useful to make a
link with Hilbert scheme techniques when the global base field is the
function field of a curve over a finite field.
