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Hamid AHMADINEZHAD - Pliability of Mori fibre spaces in dimension 3 |
| I begin with recalling the definition of pliability for Mori fibre spaces, and Fano varieties in particular. Then I explore the strong connection between this invariant and the finite generation of certain Cox rings. This, for instance, highlights the geometry behind the method of maximal singularities, used to prove birational rigidity. Some applications in proving birational rigidity (and hence non-rationality) and birational non-rigidity for some Fano threefolds will be mentioned.
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Serge CANTAT - Examples of (pseudo-)automorphisms in dimension 3 |
| I shall describe examples of complex projective threefolds with
a nice group of automorphisms or pseudo-automorphisms, the dynamics of which is non-trivial. These examples are due to Keiji Oguiso and his co-authors.
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Pierrette CASSOU-NOGUES - Field generators |
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Let k be a field. A field generator is a polynomial F∈k[X,Y] satisfying k(F,G)=k(X,Y), for some G∈k(X,Y). If G can be chosen in k[X,Y], we call F a "good field generator",
otherwise F is a "bad field generator". These notions were first studied by Abhyankar, Jan and Russell in the 70s. Until 2005, only two examples of bad field generators were known.
We study the behavior of bad field generators under birational morphisms. This lead to the construction of new bad field generators with arbitrary number of dicriticals.
Joint work with Daniel Daigle, U. Ottawa.
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Jean-Louis COLLIOT-THÉLÈNE - Unverzweigte Kohomologie |
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This talk will be part of the Perlen-Kolloquium (classical colloquium of the institute).
Die Brauergruppe von Varietäten kann man zu verschieden Zwecken benutzen :
-- Zeigen, das eine Varietät nicht rational ist, d. h., ihr Funktionenkörper nicht
rein transzendant ist.
-- Wenn der Grundkörper ein Zahlkörper ist, zeigen, daβ
das Lokal-Global Prinzip für rationale Punkte nicht gilt (Brauer-Maninsche Hindernis).
-- Die Picardgruppe, d.h. die Klassengruppe von Divisoren, untersuchen (Tatesche Vermutung).
Unverzweigte Kohomologiegruppen sind Verallgemeinerungen der Brauergruppe.
Ich werde erklären, wie man sie zu ähnlichen Zwecken anwenden kann.
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Julie DÉSERTI - On cubic birational maps of  3( ) |
| I will deal with the irreducible components of the set of birational maps of bidegree (3,3), (3,4) and (3,5).
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Alexander DUNCAN - Equivariant Unirationality |
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A variety X is unirational if there exists a dominant rational map from
an affine space V to X. I discuss a generalization of this notion which
I call ``G-unirationality.'' Consider a variety X with an action of a
linear algebraic group G. We say that X is G-unirational if there
exists an equivariant dominant rational map from a linear representation
V of G.
I will discuss how existing unirationality constructions in the
arithmetic case can be used to provide constructions in the equivariant
case. I also discuss how progress in the equivariant setting may yield
results in the arithmetic setting. The main objects of study will be
del Pezzo surfaces.
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Frédéric MANGOLTE - Topology of real algebraic varieties of dimension 3 |
| We know since Nash (1952) and Tognoli (1973) that any compact smooth manifold M admits a real algebraic model. Namely, given a manifold M, there exists polynomials with real coefficients whose locus of common real zeroes is diffeomorphic to M.
Bochnak and Kucharz proved later that there exists in fact an infinite number of distincts models for a given M. We try therefore to find "simpler" models than the others in a meaning to be specified. In this talk, I will describe the state-of-the-art concerning this research program about "simple" real algebraic models for low dimensional varieties.
The situation for curves and surfaces is quite well understood now, and the surface case is already interesting. For real algebraic threefolds, János Kollár opened in 1999 a direction of research thanks to his solution of Minimal Model Program over the reals. We will discuss several Kollár?s conjectures solved since then.
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Andriy REGETA - Lie subalgebras of vector fields and the Jacobian conjecture |
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We study Lie subalgebras L of the vector fields Vecc( 2) of the affine 2-space 2 of constant divergence, and we classify those L which are isomorphic to the Lie algebra aff2 of the group Aff2(K) of affine transformations of 2.
We then show that the following statements are equivalent:
(a) The Jacobian Conjecture holds in dimension 2;
(b) All Lie subalgebras L ⊂ Vecc(2) isomorphic to aff2 are conjugate under
Aut(2);
(c) All Lie subalgebras L ⊂ Vecc(2) isomorphic to aff2 are algebraic.
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Damiano TESTA - Unirationality of certain del Pezzo surfaces of degree two |
| A consequence of the Segre-Manin Theorem is that a del Pezzo surface of degree two is unirational over the ground field as soon as it contains a general point. I will report on joint work with C. Salgado and T. Várilly-Alvarado where we find precise conditions implying unirationality. In particular, we prove that all del Pezzo surfaces of degree two over finite fields are unirational with at most three possible exceptions.
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Algebraic and Arithmetic Geometry - "Birpol6"