Schedule
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| Saturday May 2, 2009 |
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1:00-2:00pm Mark de Cataldo (SUNY, Stonybrook) |
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2:15-3:15pm Pramath Sastry (East Carolina University) |
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4:00-5:00pm B. Purnaprajna (Kasas University, Lawrence) |
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5:15-6:15pm Izzet Coskun (University of Illinois, Chicago) |
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| Sunday May 3, 2009 |
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9:30-10:30am Dincer Guler (University of Missouri, Columbia) |
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11:00-12:00pm G.V. Ravindra (University of Missouri, St. Louis) |
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Abstracts |
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| Purna Bangere |
Title:
Canonical and Bi-canonical maps of surfaces of general type
Abstract:
Canonical map of a curve of general type is either one to one or is a finite
morphism of degree 2. In the latter case, the curve is hyperelliptic and the
image of the canonical map is a curve of minimal degree. For surfaces the
situation is far more complicated due to the existence of higher degree
covers. The canonical covers of a surface of minimal degree are ubiquitous
in the study of geometry of surfaces of general type, to name a few: they
appear in the so-called mapping geography of surfaces of general type, they
appear inevitably in the study of the canonical ring of a variety of general
type, they are unavoidable boundary cases in the study of very ampleness of
linear series on threefolds such as Calabi-Yau. The degree two canonical
covers were classified by Horikawa in the 1970's and the degree three covers
by Konno in the 1990's. In recent years, in a joint work with F. J. Gallego
degree four Galois canonical covers were classified and the classification
results show that the quadruple canonical covers are unique among canonical
covers of all other degrees from various perspectives. In this talk we will
concentrate on the the diverse behavior of the bi-canonical maps of
quadruple canonical covers, diversity unseen in the lower degree canonical
covers and more generally even in genus two fibrations, class of surfaces
that have a frequent presence in geometry of surfaces of general type. The
bicanonical maps play a prominent role in some important contexts; in a
recent work with R. V. Gurjar, it is shown that the Shafarevich conjecture
on holomorphic convexity is true for genus two fibrations and the proof
crucially depends on the nature of the bicanonical maps for genus two
fibrations. I will be speaking on these topics in my talk. |
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| Mark Andrea de Cataldo |
Title:
Filtrations in cohomology and geometry
Abstract:
I will report on joint work with L. Migliorini
where we describe the perverse filtration in cohomology using
the Lefschetz hyperplane theorem. |
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| Izzet Coskun |
Title:
Positivity in the cohomology of homogeneous varieties
Abstract:
Homogeneous varieties are central to algebraic geometry,
representation theory and combinatorics. In this talk, I will survey recent
developments in finding positive geometric rules (Littlewood-Richardson
rules) for computing the cohomology of homogeneous varieties. I will focus
on the ordinary Grassmannians, flag varieties and orthogonal Grassmannians
and flag varieties. |
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| Guler Dincer |
Title:
Non-positivity loci of linear series
Abstract:
Non-nef and non-ample loci of a divisor introduced by Nakamaye,
Lazarsfeld and others seem to promise a key for a better understanding of
linear series. In this talk we will show that if D is a big divisor then
one can explicitly describe the non-nef locus both analytically in terms
of the Lelong numbers and algebraically in terms of the multiplicity.
Some applications to Zariski type decompositions will be discussed. |
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| G.V. Ravindra |
Title:
Curves and vector bundles on threefolds
Abstract:
The Noether-Lefschetz theorem says that for a "very general"
hypersurface $X$ of degree at least 4 in P^3, any curve C in X occurs
as an intersection X\cap S where S is another surface in P^3.
Motivated by this, Griffiths and Harris asked whether any curve C in
X, where X is a general hypersurface of degree at least 6 in P^4, is
an intersection of X with a surface S in P^4. C. Voisin showed that
there exists curves which are not of this form. One would still be
interested to know whether there is indeed a "generalised
Noether-Lefschetz theorem" in this situation.
We will show that arithmetically Cohen-Macaulay (ACM) curves and
bundles provide an answer in this direction.
1) In the first part, we shall show that ACM curves provide examples
of curves which are not intersections as above and that show that
Voisin examples can be understood as special cases.
2) We will then sketch a proof of the following characterisation of
complete intersection curves on a general smooth projective
hypersurface of dimension three and degree at least six:
" a curve in such a hypersurface is a complete intersection if and
only if it is arithmetically Gorenstein (i.e. it is the zero locus of
a non-zero section of a rank two bundle with vanishing intermediate
cohomology). "
Apart from the obvious motivation for such a theorem, we shall show
a) how this theorem can be thought of as a generalisation of the classical
Noether-Lefschetz theorem for curves on two dimensional hypersurfaces.
b) that this implies the following interesting fact: a "general"
homogeneous polynomial in five variables of degree at least six cannot be
obtained as the Pfaffian (square root of the determinant) of an even sized
"minimal" skew-symmetric matrix with homogeneous polynomial entries.
c) Relate it to Horrocks' criteria for a vector bundle on projective
space to be split, and finally
d) show that it can be viewed as a verification of a (strengthening of a)
conjecture of Buchweitz-Greuel-Schreyer in commutative algebra and
algebraic geometry.
Time permitting,
3) we shall indicate its generalisation to complete intersection
subvarieties,
and finally talk about
4) an application of ACM rank 2 bundles to the study of moduli of
semi-stable bundles on cubic and quartic hypersurfaces.
The first two parts are joint work with N.Mohan Kumar and A.P.Rao, the
third is joint with J.Biswas and the fourth is joint work with
I.Biswas and J.Biswas.
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| Pramath Sastry |
Title:
Residues and Duality on formal schemes
Abstract:
There is a well known compatibility between global Grothendieck Duality and
local duality.
What was missing is a unifying viewpoint. This was provided in joint work of
Alonso, Jeremias and Lipman, where they generlize Grothendieck's work to
maps
of formal schemes which are proper modulo defining ideals of the source and
target. The classical local duality statement for (say) a smooth closed
point on
a variety then becomes Grothendieck duality for the formal scheme obtained
by
completing the variety at the closed point. Grothendieck's trace map then is
the residue on this formal scheme.
Removing the properness assumption above is non-trivial. The correct
notion of
``upper shriek" for a map of formal schemes which is of finite type modulo
defining
ideals of the source and target is not easy to come by for Nagata's
comactification theorem
(an essential ingrediant in Deligne's work on Grothendieck Duality) is not
available, and even
in the compactifiable case, it is not a piori clear that answers are
independent of
compactifications. Among other inputs is 100+ joint paper with Lipman and
Nayak, a 70+ page
paper of Nayak. The talk will survey what the broad issues were, as well
as some recent work
of mine which views some of these issues from the veiwpoint initiated by
Deligne and Verier.
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