|
Rüdiger
Achilles
|
Università
di Bologna
|
| (Aula Buzano,
Monday, April 11, 2011, 17:55–18:15) |
| Self-intersections
of
curves
and
the
degree
of
their
secant
varieties |
The talk is based
on joint work with M. Manaresi and P. Schenzel. Using
the Stückrad-Vogel
self-intersection cycle of a curve in projective
space, we obtain a formula which relates
the degree of the secant
variety, the degree and the genus of the curve as well as
the
self-intersection numbers, the multiplicities and the number of
branches of the curve at
its singular points. We deduce an expression
for the difference between the arithmetic and
geometric genus of the
curve and illustrate the usefulness of the formulas for
computer
computations.
|
|
David
Buchsbaum
|
Brandeis University
|
| (Aula Buzano,
Tuesday, April 12, 2011, 11:00–11:20) |
Greco at Brandeis
|
After
some
preliminary
remarks,
we
talk
about
Silvio
Greco's
stay
at
Brandeis.
|
|
Antonio Campillo
|
Università di
Valladolid
|
| (Aula Buzano,
Tuesday, April 12, 2011, 15:00–15:45) |
Geometry and
Poincaré Series
|
Sets
of
valuations
are
often
associated
to
algebraic-geometrical
or
analytic
varieties.
They provide filtrations by multi-index on their
rings of functions, and those filtrations
give rise, in a rather
general context, to a multi-variable Poincaré power
series.
This
Poincaré series usually encodes the information on those sets of
valuations on the
variety, and, in the most interesting cases (curves,
surfaces, non degenerated hypersurfa-
ces,...) such information is
explicitly decoded and proved to be coincident with
key
geometrical
information on the variety (Alexander polynomials, Seiberg-Witten
invariants,
zeta functions,..). When the variety is embedded and
valuations are taken in the ambient
space, two different kind of
embedded filtrations, and therefore Poincaré series,
exist;
one
geometrical, due to Ebeling and Gussein-Zade, and other algebraic, due
to Lemahieu.
Again the information can be decoded in interesting cases (curves,
Newton filtrations,
etc.). Finally, Poincaré series can
arise also
as limits in a rather general context,
and have unexpected
applications. We review the results of last ten years, originated
in
join research with Delgado and
Gussein-Zade.
|
|
Maria Virginia Catalisano
|
Università
di Genova
|
|
(Aula Buzano,
Tuesday, April 13, 2011, 10:20–10:40)
|
Higher secant varieties of
Segre and Segre-Veronese varieties
|
The problem of
determining
the dimensions of the higher secant varieties of an
embedded
projective
variety has been the object of study by many of the most outstanding
geometers
of the 19th and 20th centuries. Original investigations
mostly concentrated on the secant
line variety and were concerned with
questions of projection. By counting parameters one
finds a
natural expected dimension for these varieties, so the question
becomes: when is
the expected dimension equal to the actual dimension?
Varieties which had secant varie-
ties of less than the expected
dimension were especially interesting and were the object
of
intense study.
|
|
| Luca
Chiantini |
Università
di Siena
|
| (Aula Buzano,
Tuesday, April 12, 2011, 16:55–17:40) |
On the representation of polynomials
|
I will discuss
geometrical methods and new results for a canonical description of
gene-
ral homogeneous polynomials, from a combinatorial
point of view (determinants,fewno-
mials, pfaffians, etc.) The problem is
linked with the existence of subschemes of
some
specific types, in general
hypersurfaces.
|
|
| Ciro
Ciliberto |
Università
di Roma Tor Vergata
|
| (Aula Buzano, Tuesday, April 12, 2011,
11:30–12:20) |
Stable
Maximal
Rank
Linear
Systems
on
General
Rational
Surfaces
|
It is known that
the Segre–Harbourne–Gimigliano–Hirschowitz (SHGH) conjecture for linear
systems of plane curves with general multiple base points implies
Nagata’s conjecture.
This,in turn, can be expressed as a property of the Mori cone of the
blow–up of a plane
at ten or more general points. I propose here a rather natural
intermediate conjecture,
i.e it is implied by the SHGH and implies Nagata. I give some little
evidence for it, by
showing that the property it predicts holdsin fact for some
linear systems on the blow–up
of a plane at ten or more general points. This is based on joint work
in progress with
B. Harbourne, R. Miranda and J. Roé.
|
|
| Roberta
Di
Gennaro |
Università di Napoli
|
| (Aula Buzano, Tuesday, April 12, 2011,
09:30–09:50) |
| Liaison and
Cohen-Macaulayness conditions |
Using liaison, we
study projective curves which are close to complete intersections
in
terms of the Castelnuovo-Mumford regularity or degree and we obtain
conditions for a
space curve C being arithmentically Cohen-Macaulay, generalizing a
result of E. D.
Davis. These conditions apply also, in weaker forms, to
equidimensional and lCM sub-
schemes of codimension two and to curves in Pn, with
n >3. The results for curves are
sharp as suitable examples
show.
|
|
Monica Idà
|
| Università
di Bologna |
| (Aula Buzano, Tuesday, April 12, 2011,
11:30–12:20) |
Plane
rational
curves
and
the
splitting
of
the
tangent
bundle
|
Let
C be a rational plane curve in a projective space of given degree d
and singular
points p1,...,pn of multiplicity m1,...,mn;
it
is
natural
to
ask
how
the
pull
back
of
the tangent bundle of the
projective space splits on the normalization of C.
This
question is
strictly related to the study of the resolution of plane fat point
schemes.
I will talk about recent joint work with A.Gimigliano and B.Harbourne
on this subject.
Assuming that the points p1,...,pn are generic,
and
that the proper transform D of C
in the blow up of the plane at p1,...,pn is
smooth, we give a complete determination
of the
splitting types when n is less or equal 7. The case that D^2=-1 is of particu-
lar interest. For n less or equal
8, there are only finitely many exceptional curves,
and all of them
have balanced splitting. However, for n=9 there are infinitely
many
exceptional curves having unbalanced splitting. These new examples are
related to a
semi-adjoint formula which we conjecture accounts for all
occurrences of unbalanced
splitting when $D^2=-1$ in the case of nine
generic points.
|
|
Salvatore
Giuffrida
|
Università
di Catania
|
|
(Aula
Buzano,
Wednesday,
April
13,
2011,
10:25–10:45)
|
Scheme
theoretic
complete intersections in P1xP1
|
The
zero–dimensional
subschemes of Q =
P1
×P1 arising as
scheme–theoretic complete
inter-
sections of two curves are considered. The main goal is to
describe the possible Hilbert
functions and to give some information on
the graded Betti numbers of such schemes. In
some case the graded Betti
numbers of such subschemes is determined.
|
|
| Rosa Maria
Mirò Roig |
| Universitat de
Barcelona |
| (Aula Buzano,
Monday, April 11, 2011, 16:00–16:45) |
| The
Minimal
Resolution
Conjecture
for
Points
on
a
del
Pezzo
Surface |
It is a
long-standing problem in Algebraic geometry to determine the Hilbert
function of
any set Z of distinct points
on any projective variety of Pn.
It
is
well-known
that
HZ(t)≤ min{HX(t),|Z|} for any t, and that
the equality holds if the points are general.
A much more subtle question is to find out the exact shape of the
minimal free resolu-
tion of IZ. Mustata
conjectured that the graded Betti numbers had to be as small as possi
ble (when X =Pn, we
recover Lorenzini’s conjecture.
In my talk, I will give a brief account of the known results around
Mustata’s conjecture
and prove it for points on an ACM quasi minimal
surface.
|
|
Roberto
Notari
|
Politecnico di Milano
|
| (Aula Buzano, Tuesday,
April 12, 2011, 10:35–10:55) |
On some minimal
curves in P3
|
Biliaison
theory
is
a
very
powerful
tool for studying algebraic curves in the
projective
space P3=Proj(R =
K[x,y,z,w]), with K an algebraically closed field of characteristic 0.
In particular, every curve in the biliaison class associated to an
Artinian graded R-modu-
le M can be obtained from the minimal curves in the class with a
sequence of
ascending
elementary links. We investigate the geometrical properties of the
minimal curves
in the
case M is Gorenstein
of the form R/I, for a suitable homogeneous ideal
I.
|
|
Maria Luisa Spreafico
|
Politecnico di Torino
|
| (Aula Buzano, Tuesday,
April 12, 2011, 09:55–10:15) |
A construction of ACM curves
|
|
We describe a construction
of ACM curves in P3 and we
apply it to find ACM curves
through a given 0-dimensional
scheme.
|
|
Carlo
Traverso
|
Università
di Pisa
|
| (Aula Buzano,
Tuesday, April 12, 2011, 15:55–16:45) |
Multivariate
algebra,
integer
lattices
and
error-correcting
codes:
An
algebraist's
Adventures
in
Post-Quantum
Cryptography
|
ABSTRACT: I will explain
how, as a Groebner fan, I became curious of a challenge of the
Spectre, and contributed to defeat a quantum threat to destroy the Internet.
TRANSLATION: I will explain my
recent research that combines Groebner bases for binomial
ideals, integer lattices, and error correction with secret erasures,
developing new
variants of lattice-based
cryptosystems.
These,
differently
from
protocols
based
on
factorization,
discrete
logarithms and
ellip-
tic curves, are not broken in polymonmial time by Schor quantum
algorithms, and can
hence be used to develop signature algorithms that
will resist to the development of
efficient quantum computers.
|
|
| Angelo Vistoli |
Scuola Normale
Superiore, Pisa
|
| (Aula Buzano,
Wednesday, April 13, 2011, 09:30–10:15) |
Essential
Dimension
|
The concept of
essential dimension has been introduced 15 years ago, and has attracted
a
lot of
attention since then. The essential dimension of an algebraic or
algebro-geometric
object (e.g., of an algebra, a quadratic form, or an algebraic curve)
is the minimal
number of independent parameters required to define the underlying
structure. In many
cases computing the essential dimension is a delicate question, linked
with long-standing
open problems. I will survey the basic concepts, give some examples,
and present recent
results due to Reichstein and myself on essential dimension of
homogeneous forms.
|
|
|
