The seminar takes place every Tuesday from 1pm to 2pm in room B217 of the science building.
18.2
Olivier Benoist (CNRS-ENS)
On the rationality of real conic bundles
Deciding whether a given algebraic variety is rational (birational to projective space) is an important problem in algebraic geometry. Over the field of real numbers, this problem is particularly interesting for varieties that are known to be rational over the complex numbers, as it then has an arithmetic flavour. In this talk, I will review the techniques that are available, and focus on concrete examples of real conic bundles for which I will provide new positive and negative results. This is joint work with Alena Pirutka.
4.3
Gebhard Martin (Bonn)
Classification of non-F-split del Pezzo surfaces
Among all smooth cubic surfaces in P^3, there is a unique one without three lines forming a triangle: The Fermat cubic in characteristic p = 2. Coincidentally, this is also the unique non-F-split smooth cubic surface. By work of Hara, non-F-split del Pezzo surfaces exist only in degrees 3, 2, and 1. After describing the classification of non-F-split del Pezzo surfaces of degree 2 due to Saito, I will report on joint work with Réka Wagener in which we give a geometric characterization of non-F-split del Pezzo surfaces of degree 1.
1.4
Alain Valette (Neuchâtel)
Reciprocal hyperbolic elements in PSL_2(\Z)
An element A in PSL_2(\Z) is hyperbolic if |Tr(A)|>2. The maximal virtually abelian subgroup of PSL2(\Z) containing A is either infinite cyclic or infinite dihedral; say that A is reciprocal if the second case happens (A is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in PSL_2(\Z) in terms of the continued fractions of their fixed points in P^1(\R) (those are quadratic surds). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke & Klein.
8.4
Lukas Lewark (ETHZ)
The joy of not being a PID
Two knots (circles in 3-space) are called concordant if they form the boundary of a cylinder in 4-space. Concordance classes of knots constitute an abelian group C called the concordance group, which has been one major focus of knot theory since the 1960s. In this talk, we will construct a homomorphism (based on Khovanov homology) from C to another abelian group G. This latter group G consists of chain complexes (with coefficients in a ring R) modulo a certain equivalence relation. With R a PID, G is just infinite cyclic; but with R a non-PID, G may have infinite rank, and so – oh joy – the homomorphism from C to G may be richer in information about C.
This talk will be aimed at an audience with no background knowledge in knot theory and Khovanov homology.
6.5 Marco Golla (Nantes)
Alexander polynomials and symplectic curves in CP^2
Libgober defined the Alexander polynomial of a (complex) plane
projective curve and showed that it detects some Zariski pairs of
curves: these are curves with the same degree and the same singularities
but with non-homeomorphic complements. He also proved that the Alexander
polynomial of a curve divides the Alexander polynomial of its link at
infinity and the product of Alexander polynomials of the links of its
singularities. We extend Libgober’s definition to the symplectic case
and prove that the divisibility relations also hold in this context.
This is joint work with Hanine Awada.
20.5
Sokratis Zikas (IMPA)
On Mori Dreamness of blowups along space curves
Mori Dream Spaces are a special kind of varieties introduced by Hu and Keel that exhibit an ideal behaviour from a birational point of view. While Mori Dreamness is a very desirable property, generally, it is not preserved by even the simplest birational maps: blowups. In this talk we study Mori Dreamness of blowups along space curves: we provide various sufficient criteria as well obstructions to the blowup being a Mori Dream Space. We also study how this property behaves while varying the curve in the corresponding Hilbert scheme and show that it is neither an open nor a closed condition. Furthermore we exhibit examples of Hilbert schemes whose general element does not give rise to a Mori Dream Space, while special elements do, and vice versa.
This is joint work in progress with Tiago Duarte Guerreiro
