C-sets in Geometry and Analysis
by Dubson, Alberto S.
Whitney estimates and area properties of C-sets.
Applications to problems in Geometry and Analysis.
Non-commutative P^2 and Quiver varieties
by Ginzburg, Victor
Introduction to dual varieties and differential-geometric techniques
by Landsberg, Joseph
I will explain how to use techniques of local differential geometry
to study questions about subvarieties of projective space, with an
emphasis on dual varieties. Dual varieties are the geometric analogs
of the Legendre transform. They also encompass the discriminants, resultants,
and multidimensional determinants studied by Gelfand et. al. The study
of dual varieties is related to several areas of algebraic geometry. I will
explain connections with singularity theory and the study of vector bundles on
projective space.
Deligne dimension formulas and Vogel's universal Lie algebra viewed from the perspective of algebraic geometry
by Landsberg, Joseph
This is a joint work with L. Manivel.
With the help of a computer, Deligne, Cohen, and de Man proved
some striking dimension and decomposition formulas for the exceptional Lie algebras.
Vogel has conjectured a "universal Lie algebra" and proved even more striking dimension
and decomposition fomulas for all Lie algebras.
I will present a proof and extension of the Deligne formulas using simple geometric models
and give some geometric interpretations of Vogel's universal Lie algebra coming from
elementary projective geometry.
Stratification of the variety of Lie algebras via the moment map
by Lauret, Jorge
The space of all $n$-dimensional complex Lie algebras can be naturally
identified with the set $\lca\subset V_n$ of all Lie brackets, where
$V_n=\Lambda^2(\CC^n)^*\otimes\CC^n$. $\lca$ is an algebraic set,
since the Jacobi identity is given by polynomial conditions.
We consider the moment map $m:\PP\lca\longrightarrow\im\ug(n)$ associated
with the natural $\Gl(n)$-action on $\PP\lca$, where $\im\ug(n)\subset\glg(n)$
is the space of hermitian maps. In order to understand the stratification
of $\PP\lca$ defined by the negative gradient flow of the functional
$F=||m||^2$, we study the critical points of $F$. We obtain a description
of them in terms of those which are nilpotent, as well as many compatibility
properties between a Lie bracket which is a critical point and the
invariant inner product we have fixed on $\CC^n$ to define the moment
map. We also prove that the minima of $F:\PP\lca\longrightarrow\RR$
are essentially the semisimple Lie algebras, and the maximum is attained
only in the direct sum of the $3$-dimensional Heisenberg Lie algebra
and an abelian factor.
As an application, we study rigid Lie algebras, that is, those for
which its $Gl(n)$-orbit is open. We give a criterion for the rigidity
of a solvable Lie algebra whose nilradical is a critical point of
$F$. Such a criterion is related with the symplectic reduction, or
equivalently, with the Mumford quotient, of certain action of a reductive
subgroup of $\Gl(n)$ on an algebraic subvariety of $\PP\lca$. This
technique gives rise new examples of rigid Lie algebras and provides
alternative proofs for the rigidity of several already known examples.
Classifying projective varieties via their hyperplane sections, I and II
by López, Angelo
A classical tool to study projective algebraic varieties is recognizing
the properties of their hyperplane sections, the goal being that the more one
knows about the hyperplane section the better one knows the variety.
Recently the perspective has been reversed and several geometers have studied
when, given a projective variety $X \subset P^N$, there exists another
variety $Y \subset P^{N+1}$, different from a cone over $X$ and such that
$X = Y \cap P^N$.
We will first give a general review of the methods used to study this problem
both for curves and surfaces, mainly Zak's theorem and Wahl/Gaussian maps.
In the second part we will apply the methods to recover the celebrated
classification of Fano threefolds and we will report on recent results about
the existence of Enriques-Fano threefolds.
Frobenius splitting of reductive embeddings
by Rittatore, Álvaro
Let $G$ be a complex connected reductive algebraic group, and $X$ a
complex $G$-variety. Brion and Inamdar had shown that if $p$ is big
enough, then the reduction mod $p$ of $X$ is Frobenuis split, and in
particular has rational singularities. In this talk we will prove that
if $G$ is a connnected reductive group over a field of arbitrary
characterictic, then any $G$-embedding is Frobenuis split. As an
application we will show that the algebra of regular functions of a
connected reductive algebraic monoid $M$ has a good filtration for the
action of the unit group of $M$.
Curves of degree 5 and genus 2
by Vainsencher, Israel
Let Hilb$^{5t-1}(\p3)$ be the Hilbert scheme of closed 1-dimensional
subschemes of degree 5 and arithmetic genus 2 in $\p3$. Let H
be the component of ${\rm Hilb}^{5t-1}(P^3)$ whose generic point corresponds
to a smooth irreducible curve. We follow the footsteps of \cite{Rojas-Vainsencher}
and \cite{Vainsencher-Xavier} to construct an explicit desingularization
of H. It is suitable for enumerative applications via Bott's residue.
The idea is to use an elementary linkage argument. Let the Curve
$C\subset P^3$ correspond to a general point of H. It lies on a
unique quadric surface $f_2$. Identifying $f_2 \congP^1\times P^1$,
the curve $C$ can be seen as a curve of bidegree (2,3). Thus, it
possesses $\infty^1$ trisecant lines, all in the System (1,0). Let
$L$ be a trisecant line. Then $L\cup C$ is a curve of bidegree (3,3).
Hence, this union is a complete intersection of the quadric $f_2$
with a cubic surface $f_3\supset L$. We revert this construction,
forming for each line $L$ the family of intersections quadric-cubic
through $L$. One then proceeds to identify explicit blowup centers
in order to flatten the family.
Sesiones 1ra semana
Sesiones 2da semana