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Picard Maximal Varieties
Donu Arapura, Purdue University
Abstract:
Let me say that a smooth complex projective variety is
Picard maximal if the space of algebraic cycles in cohomology is
as large as possible, i.e. if it generates the $(p,p)$ part of the
Hodge structure for each $p$. Part of the motivation for considering
these is that various conjectures (Hodge, Grothendieck, Tate) hold
automatically for such varieties, but I also think that their study has
intrinsic value. I will briefly describe what is known in dimension
2, thanks to the efforts of Shioda and others. Then I will
look at higher dimensional examples. The main result
is that the Hilbert scheme of points on a Picard maximal
surface is again Picard maximal.
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Asymptotic Growth of Saturated Powers of Ideals and Epsilon
Multiplicity
Dale Cutkosky, University of Missouri
Abstract:
We study the growth of saturated powers of ideals and modules.
There are examples showing that the algebra of saturated powers of an ideal I
in a noetherian local ring R is not a finitely generated R-algebra;
As such, it cannot be expected that the ``Hilbert function'',
giving the length of the R-module (I^k)^{sat}/I^k, is very well behaved
for large k.
However, it can be shown that it is bounded above by a polynomial
in k of degree d, where d is the dimension of R.
We show that for quite general domains, there is a reasonable asymptotic
behavior
of this length. We extend this to the case of modules to show that the epsilon
multiplicity, defined by Ulrich, Validashti and Kleiman, exists as a limit
over very general domains.
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Some homological conjectures revisited
Hailong Dao, University of Kansas
Abstract:
The famous homological conjectures were formulated in the 50s and 60s and
have had
significant impacts on commutative algebra. Most of them can be phrased as
surprising statements about finitely generated modules with finite
projective dimension.
In this talk we will survey some new statements that sound quite similar to
the
classical conjectures, but have distinctly different features and motivations.
One notable difference is the switch from finite projective dimension property
to the property that the classes of certain modules are 0 in the rational
Grothendieck/Chow groups. We will discuss recently discovered connections to
algebraic geometry and noncommutative geometry.
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A finiteness condition on local cohomology in positive characteristic
Florian Enescu, Georgia State University
Abstract:
For a local ring R of positive prime characteristic,
the local cohomology modules of the ring with support in the maximal ideal
inherit a natural Frobenius action. The talk with discuss the lattice of
submodules of the local cohomology that are compatible with the Frobenius action.
It is known that this lattice contains remarkable information about the structure
of the ring R. A conjecture regarding it will be formulated, and recent progress on
it will be presented.
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Arithmetically Cohen-Macaulay bundles on hypersurfaces
Mohan Kumar, Washington University
Abstract:
The talk is based on some joint work with A. P. Rao and G. V. Ravindra. The
relevant preprints are available at arXiv:math.AG/0507161 ,
arXiv:math/0611620 and arXiv:math/1005.3990.
The first appeared in Commentari Math. Helv., the
second in IMRN and the third in Fields Inst. Comm.
We will also touch on some work in progress.
A vector bundle on a polarized projective variety (X;L) is called
Arithmetically Cohen-Macaulay if all its middle cohomologies in all twists
by powers of L vanish. A famous criterion of G. Horrocks states that a
vector bundle on projective space is a direct sum of line bundles if and
only if it is arithmetically Cohen-Macaulay (with respect to the usual
polarization). It is well known that this criterion fails for other
varieties, in particular for hypersurfaces in projective spaces. In my talk
I will discuss the following results proved in the above articles. Any rank two
arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree
at least three in P5 or on a general hypersurface of
degree at least six in P4 must be split.
It is also known that for general quintic threefolds, rank two ACM bundles are
rigid. So it leads to an interesting enumerative problem of counting these
(upto twists). I will discuss what is known and what is still to be done.
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Diagonal F-threshold, Hilbert-Kunz multiplicity and socle degrees of
Frobenius powers
Jinjia Li, University of Louisville
Abstract:
Let (R,m) be a local ring in prime characteristics.
The diagonal F-threshold (i.e, the F-threshold of m with respect to an
m-primary ideal I) is known to exists as a real number in either the
F-pure on the punctured spectrum case or
in the standard-graded complete intersection case.
In this talk, we make connections between the rationality of this
invariant with that
of the Hilbert-Kunz multiplicity by investigating the socle degrees
distribution of the Artinian quotient of R (modulo the Frobenius
powers).
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Some new results on local cohomology
Gennady Lyubeznik, University of Minnesota
Abstract:
We will describe some striking recent results of our student Yi Zhang
on local cohomology modules of polynomial rings in finitely many variables
over a a field of characteristic p>0. Proofs, not surprisingly,
involve the Frobenius morphism. There is no doubt that characteristic
zero analogues of these results are true but proofs are yet to be found.
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F-signature and F-rational signature
Mel Hochster, University of Michigan
Abstract:
After first surveying what is known about F-signature, which is a
subtle invariant of local rings (R,m) of prime characteristic p > 0
that gives some measure of how singular they are, the talk will
discuss recent results of Yongwei Yao and the speaker concerning
a new notion, F-rational signature. Positivity of F-signature
characterizes strongly F-regular rings, while positivity of F-rational
signature characterizes F-rational rings. One key point is that if I is
an m-primary ideal there is a positive real constant c(I) such that for
all ideals J between I and m, the difference of the Hilbert-Kunz
multiplicities of R with respect to I and J, if nonzero, is bounded
below by c. A number of open questions will be discussed.
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Geometry of surfaces of general type
Bangere Purnaprajna, University of Kansas
Abstract:
In this talk I will survey my recent results with my coauthors on
varieties of general type with particular emphasis on
the case of algebraic surfaces. The first theme will relate the
deformation of canonical maps to construction of varieties of general
type with prescribed invariants. The framework we develop
allows us to describe some components of infinitely many moduli
spaces of surfaces of general type.
The second theme is to explore a higher dimensional analogue
of the uniformization theorem of
Riemann and Kobe, the so-called holomorphic convexity of the universal
cover of a projective variety, which goes under the name of Shafarevich
conjecture. Until recently, this was not known in its full generality
for even surfaces fibered by genus two curves. We prove some general
statements about fundamental groups of surfaces fibered by hyperelliptic
curves of arbitrary genus. Examples show that this is an optimal result.
As a byproduct we prove, a stronger form of Shafarevich conjecture for
these surfaces, and a very attractive conjecture of Nori on fundamental
groups. This also yields statements on second homotopy groups of fibered
surfaces.
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Local theory of self maps,
Lucien Szpiro, City University of New York
Abstract:
We will report on work with Mahdi Majidi Zolbanin and Nikita Miasnikov
concerning self maps of local rings:
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Extension theorems a la Fakhruddin;
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Regularity criterium a la Kunz (and Avramov, Miller, and Iyengar);
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Existence of entropy a la Samuel.
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Multiplicities and equisingularity
Bernd Ulrich, Purdue University
Abstract:
This talk will give a general survey of the connection between
equisingularity theory and multiplicities. The last part of the lecture
will be devoted to more recent results and a new notion of multiplicity,
the ε-multiplicity.
One of the goals in equisingularity theory is to devise criteria for
analytic sets to be 'alike', most notably when these sets occur in a family.
Ideally, such criteria only depend on numerical information about the
individual members rather than the total space of the family. The numerical
invariants often used are suitably dened multiplicities. Many
of the known equisingularity conditions, such as Whitney's condition
B or Verdier's condition W, involve limits of tangent spaces, and it was
Teissier's seminal insight to relate these convergence properties to the
purely algebraic concept of integral dependence of ideals. Thus he was able
to show that a family of isolated hypersurface singularities is
Whitney equisingular if and only if the Hilbert-Samuel multiplicity
of certain Jacobian-like ideals is constant across the family. Any
generalization beyond the case of isolated hypersurface singularities however,
necessitates the use of Jacobian modules rather than ideals and requires new
notions of multiplicities. The last part of the talk will survey
such generalizations, including recent work with S. Kleiman and J. Validashti,
were families of arbitrary isolated singularities are treated.
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