Abstracts of the talks at caag6 at chennai
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R. Sujatha
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Title: "Fine Selmer groups for elliptic curves"
Abstract: We study the fine Selmer groups for elliptic curves and
discuss a conjecture formulated for these groups. This is joint work
with J. Coates.
Jaya Iyer
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Title: "Chern classes of flat bundles"
Abstract: We discuss the primary invariants of a flat bundle on a
variety, in various cohomology theories. We give examples of flat
bundles (some Gauss-Manin systems) having trivial Chern classes
in the rational Chow groups and some flat bundles having
non-trivial Chow Chern classes.
R. Parthasarthi
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Title: "Donaldson Uhlenbeck compactification of parabolic bundles"
Abstract: We construct the algebro-geometric Donaldson-Uhlenbeck
compactification of the moduli space of semistable
parabolic bundles over smooth connected projective surface
with irreducible parabolic divisor D. We also
construct a stable parabolic bundle of any rank.
Good reduction, bad reduction
Manoj Kumar Keshari
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Title: "A note on projective modules over real affine algebras".
Abstract:Abstract: Let $A$ be an affine algebra over the field of real
numbers $\mathbb R$ of dimension $d$. Let $f\in A$ be an element not
belonging to any real maximal ideal of $A$. Let $P$ be a projective
$A$-module of rank $\geq d-1$. Let $(a,p)\in A_{f}\oplus P_f$ be a
unimodular element. Then, the projective $A_{f}$-module $Q=A_{f}\oplus
P_f/(a,p)A_{f}$ is extended from $A$.
Pooja Singla
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Title:"The regularity of power products of graded
ideals and minimal monomial reduction ideal"
Abstract:In this lecture, we discuss the regularity of the power
products $I_1^{\ell_1}I_2^{\ell_2}\cdots I_m^{\ell_m}$
It is known that for large enough $\ell_i$ the regularity function
for these power products is a multi-linear function of the form
$\sum_{i=1}^mp_i\ell_i+c$. We determine the coefficient $p_i$ of
this function. In case of monomial ideals, we give a convex
geometric interpretation of the $p_i$. In fact we show that these
coefficients are determined by the minimal monomial reduction
ideals of the factors $I_j$$ Quite generally we show that any
monomial ideal $I$ has a unique minimal monomial reduction ideal
and that it is determined by the convex hull of $I$ and its
extremal points. We also give a convex geometric proof of the fact
that the reduction number of a monomial ideal with respect to the
minimal monomial reduction is uniformly bounded. This is a special
case of the Generaliz$ theorem by B.Johnston and D.Katz for graded
ideals.
Gurjar. R
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Title:"The depth of $\Omega^1$ for certain local rings"
Abstract: Let R=$T/P$ be a geometric local ring of dimension
$>2$, where $T$ is a geometric regular local ring with algebraically
closed residue field of char. $0$ and $P$ is a prime
ideal in $T$. If $R$ satisfies Serre's condition $S_2$ and depth
$T/P^{(2)}\geq 2$, where $P^{(2)}$ is the second symbolic power of $P$,
then depth $\Omega^1_R>0$.\\
This is a joint work with Vinay Wagh.
G.Kemper
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Title:"Depth of invariant rings and wild ramification"
ABSTRACT: The depth of an invariant ring R^G provides a nice measure
for its homological complexity. In this talk we consider the case that
R is a Cohen-Macaulay ring and the group G has order divisible by the
characteristic of R. By relating the depth of R^G to group cohomology
we obtain upper bounds for the depth in terms of the wild ramification
locus of the G-action. This is joint work with Nikolai Gordeev.
P. Russell
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Title:"Some results on affine rational surfaces"
Abstract: An affine rational surface has "trivial Makar-Limanov
invariant" if it admits "many" actions by the additive group.
I will outline a complete description of normal surfaces
with this property in case the Picard rank is 0. I will
discuss some examples and questions for higher Picard rank.
V.B. Mehta
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Title:"Fundamental Group Scheme in char p"
Abstract:
Alok,K.Maloo
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Title:"Maximally differential ideals in positive characteristic"
Abstract:Let $A$ be a ring and let $\c D$ be a set of
derivations of $A$. An ideal $I$ of $A$ is said to be $\c
D$-differential if $d(I)\subset I$ for all $d\in \c D$ and
maximally $\c D$-differential if $I$ is $\c D$-differential,
proper and it is maximal with respect to these properties.
Let $I$ be a maximally $\c D$-differential ideal of $A$. Study of
the structure of $I$ has been an interesting topic of research for
quite sometime. Indeed, many of the results describe the structure
of $I$ completely in various types of rings, e.g., Balwant Singh,
in a complete Noetherian local ring containing a field of
characteristic zero, Y. Ishibashi, in a Noetherian graded ring
whose degree zero elements form a field of characteristic zero
and the author, in a Noetherian local ring containing a field of
positive characteristic.
The present talk is based on some recent results obtained by the
author in a graded ring $A=\oplus_{i=0}^\infty A_i$, where $A_0$
is a field of positive characteristic. The main result of the talk
is\\
\noindent Let $A = \oplus _{i=0}^{\infty}{A_i}$ be a Noetherian
graded ring such that ${A_0} $ is a field of characteristic $p>0$.
Let $\c D$ be a set of derivations of $A$. Let $I$ be a graded
ideal of $A$ and $r=\emdim(A/I)$. Then the following are
equivalent
\begin{enumerate}
\item[{\rm (a)}] $I$ is maximal among proper $\c D$-differential
graded ideals.
\item[{\rm (b)}] There exist a Noetherian graded subring $B$ of
$A$ and homogeneous elements $x_1,x_2,\ldots,x_r$ of $A$ such
that $\{x_1,x_2,\ldots, x_r \}$ forms a $p$-basis of $A$ over
$B$ and $I = {\mathfrak n}A$, where ${\mathfrak n}$ is the
irrelevant maximal ideal of $B$.
\end{enumerate}
Shyamashree Upadhyay
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Title:"Hilbert Functions of points on Schubert verieties
in the Orthogonal Grassmannian"
Abstract:Fix a vector space \textbf{V} of finite
dimension over an algeraically closed field of characteristic
not equal to 2 and a non-degenarate symmetric bilinear form
on it.The set of all \textit{maximal isotropic subspaces} of
\textbf{V} \(those\ on\ which\ the\ bilinear\ form\
vanishes\) is called the \textit{Orthogonal Grassmannian}.It
has naturally the structure of a smooth projective variety.
We compute the multiplicity and the Hilbert function of the
local ring at any given point of a Schubert sub-variety of
the \textit{Orthogonal Grassmannian}.
Tony Joseph, P
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Title:"On a filtration of the canonical module"
Abstract: Let $(A,\m)$ be a complete Cohen-Macaulay local
ring with canonical module $\omega$. Let $I$ be an
$\m$-primary ideal of $A$ such that the associated graded
ring $G_I(A)$ is Cohen-Macaulay. We prove that there exists
(essentially unique) $I$-stable filration $F$ on $\omega$
such that $G(F)$ (the associated graded module of the
filtration) is the canonical module of $G_I(A)$.
Trivedi Vijayalaxmi
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Title:"Hilbert-Knuz multiplicities and vector bundles on curves"
Abstract: Here we relate semistablity and Frobenius semistablity
property of a given vector bundle on a projective curve with HK
multiplicity of the associated standard graded ring.
As a consequence we prove that the HK mutilplicity of a two
dimensional standard graded ring is rational. In the case of plane
curves we give a numerical characterization of semistability of the
kernel bundle under the Frobenius map via HK multiplicity.
Moreover, we show that the HK multiplicities of the reductions to
positive characteristics of an irreducible projective curve in
characteristic 0 have a well defined limit as the characteristic
tends to infinity.
Parimal, R
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Title:"Division algebras of prime degree over function fields
of surfaces"
Abstract:
C.S. Dalawat
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Title:"Good reduction, bad reduction"
Abstract: This expository talk will be devoted to criteria for good
reduction of varieties over local fields. We shall review old and
new results on the good reduction of curves, abelian varieties and
twisted forms of projective spaces.
Vinay Wagh
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Title:"Some results on the modules of derivations of certain
local rings"
Abstract:Here we give a bound on the multiplicity of
$n$-dimensional rings of invariants and a bound on the minimum
number of generators of derivation modules of two dimensional
rings of invariants. We also give the explicit generators for
the ring of invariant of cyclic diagonal subgroup of
$GL_2(\mathbb{C})$.
Selby, Jose
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Title:"The unimodular vector quotient is nilpotent"
Abstract:A. Suslin has defined the Suslin matrices
$S_r(v,w) \in Gl_{2^r}(R)$,where $v,w \in Um_{r+1}(R)$
with $ = 1$. Let $SUm_r(R)$ dente the subgroup of
$Gl_{2^r}(R)$ generated by the Suslin matrices. We define
a homomorphism $\varphi : SUm_r(R) \to SO_{2(r+1)}(R)$
which leads to an injective map, for $r > 1$,
$\frac{SUm_r(R)}{EUm_r(R)} \to
\frac{SO_{2(r+1)}(R)}{EO_{2(r+1)}(R)}$,
where $EUm_r(R)$ is the
elementary subgroup of $SUm_r(R)$. Using this and ideas
of A. Bak we prove that $SUm_r(R)/EUm_r(R)$ is nilpotent.
S.Ramanan
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Title:"Fano Varieties"
Abstract:
Avinash Sathaye
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Title:"Globalization of an old Theorem of Zariski"
Abstract:I will describe the joint work with Abdallah Assi
(On Quasihomogeneous Curves - preprint).
Let the ground field be algebraically closed of characteristic zero.
In a short paper (1966: Collected Works V.3, p. 475-480), Zariski
characterized plane unibranch curves having maximum torsion, to be
exactly curves of the form y^a-x^b with a,b coprime, after a suitable
local change of variables . This torsion came out to be the length of
the module of differentials of the integral closure modulo the module
of differentials of the original coordinate ring
(\Omega(\bar{R})/(\Omega(R)).
We show that this concept can be defined similarly for an affine curve
with one place at infinity and prove a similar characterization,
namely the length is maximal if and only if the curve is of
Lin-Zaidenberg type, meaning after a suitable change of coordinates,
it is of the form y^a-x^b with a,b coprime.
The well known Abhyankar Moh theory of plane affine curves with one
place at infinity gives necessary conditions for such a curve to be
rational, but the characterization of such curves has not been known.
The relative module of differentials gives a new tool to study
additional conditions on a rational plane curve with one place at
infinity .
We will describe additional calculations of this relative module of
differentials.
V.Balaji
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Title:"Principal bundles on algebraic surfaces"
Abstract:
V.Suresh
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Title:
Abstract:
T.E.Venkata Balaji
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Title:"The Witt-Invariant Classifies Ternary Quadratic Bundles"
Abstract: We study degenerations of rank 3 quadratic forms using those
of rank 4 Azumaya algebras, and extend what is known for good forms and
Azumaya algebras. By considering line-bundle-valued forms, we extend
the theorem of Max-Albert Knus that the
Witt-invariant---the even-Clifford
algebra of a form---suffices for classification. The general, usual and
special orthogonal groups of a form are determined in terms of
automorphism groups of its Wittinvariant. Martin Kneser's
characteristic-free notion of semiregular form is used. Examples of
non-existence of good forms and Azumaya structures are given.
J.K. Verma
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Title:"Hilbert coefficients and depth of fiber cones of ideals"
Abstract:Criteria are given in terms of certain Hilbert
coefficients for the fiber cone F(I) of an m-primary ideal I in a
Cohen-Macaulay local ring (R,m) so that it is Cohen-Macaulay
or has depth at least dim(R)-1. A version of Huneke's
fundamental lemma is proved for fiber cones. S. Goto's results
concerning Cohen-Macaulay fiber cones of ideals with minimal
multiplicity are obtained as consequences.
(This is a joint work with A. V. Jayanthan)