Seminar Algebra and Geometry - Fall 2022

It happens on Tuesdays from 10:30 to 12:00 in Spiegelgasse 1, Seminarraum 00.003 (ground floor). Some tea takes place before the talk, at 10:00 in Spiegelgasse 1, 6th floor.
September 20 Marc Abboud
Dynamical degrees of endomorphisms of complex affine surfaces are quadratic integers Let X0 be a complex affine surface and let f be a dominant endomorphism of X0. The dynamical degree λ1 of f is defined as follows. Let X be a projective surface with an embedding ι: X0X such that ι(X0) is an open dense subset of X. Take any ample divisor H on X and define λ1 := limn → ∞ ( ( fn )* H \cdot H )1/n. This does not depends on H nor on the projective surface X; one has λ12 ≥ e where e is the topological degree of f. In 2007, Favre and Jonsson showed when X0 is the complex affine plane that the dynamical degree of any endomorphism is a quadratic integer. We extend their result to the case of any complex affine surface. The proof uses the space of valuations centered at infinity V. The endomorphism f defines a transformation of V and studying the dynamics of f on V gives information about the dynamics of f on X0. The main result is that when λ12 > e, f admits an attracting fixed point in V that we call an eigenvaluation. This implies that one can find a good compactification X of X0 such that f admits an attracting fixed point p at infinity and f has a normal form at p; the result on the dynamical degree follows from the normal form.
September 27 Gene Freudenburg
Western Michigan University
Actions of SL2(k) on affine k-domains via fundamental pairs Let k be a field of characteristic zero. In their 2012 paper, Arzhantsev and Liendo write:
" A regular SL2-action on an k-affine variety X is uniquely defined by an 𝖘𝔩2-triple { ∂, ∂+,∂-} of derivations of the algebra k[X], where the ∂± are locally nilpotent, ∂ =[∂+,∂-] is semisimple, and [∂,∂±]=± 2∂±."
Accordingly, we define a fundamental pair of locally nilpotent derivations of the affine k-domain B as a pair (D,U) which satisfies
  1. [D,[D,U]]=-2D and [U,[D,U]]=2U, and
  2. B=⊕d∈ℤBd is a ℤ-grading where Bd=ker ([D,U]-dI).
Let AB be the kernel of D. Classical techniques show that A is affine. The Structure Theorem describes A as an ℕ-graded ring, as well as the degree modules and image ideals of D. The Structure Theorem is used to classify normal affine SL2(k)-surfaces with trivial units, generalizing the classification given by Gizatullin and Popov for complex SL2(ℂ )-surfaces. This classification is used to describe three-dimensional UFDs which admit a certain kind of SL2(k) action. From this, we show that any SL2(k)-action on 𝔸k3 is linearlizable, which was proved by Kraft and Popov in the case k is algebraically closed. This description is also used, together with Panyushev's theorem on linearization of SL2(k)-actions on 𝔸k4, to show a cancelation property for threefolds X:
Let k be an algebraically closed field of characteristic zero, and let X be a threefold such that Xₓ 𝔸k1\cong𝔸k4. If X admits a nontrivial action of SL2(k), then X is isomorphic to 𝔸k3. We close with several open questions about SL2-actions.
October 11 Marta Benozzo
University College London
Iitaka conjecture for anticanonical divisors in positive characteristic Given a fibration f : XZ, a natural question is how we can relate the Kodaira dimensions of X, Z, and the fibres. The Iitaka conjecture addresses this problem with an inequality. In characteristic 0, it is proven in many cases, while it is known to not hold in general in positive characteristics. Recently, a similar statement for the anticanonical divisors was proven in characteristic 0. The same results can be extended also to positive characteristics in low dimensions.
October 18 Gebhard Martin
Automorphisms of del Pezzo surfaces in positive characteristic One of the key ingredients in the classification of conjugacy classes of finite subgroups of the complex plane Cremona group is understanding the automorphism groups of conic bundles and del Pezzo surfaces. While automorphisms of del Pezzo surfaces of degree at least 5 behave uniformly in all characteristics, this is no longer true in smaller degrees. Nevertheless, a complete classification of automorphism groups of del Pezzo surfaces of degree 4 and 3 in positive characteristic has recently been obtained by Dolgachev and Duncan. In this talk, I will report on ongoing joint work with Dolgachev on the classification of automorphism groups of del Pezzo surfaces of the remaining degrees 2 and 1.
November 1 Mani Esna-Ashari
University of Basel
tba tba
November 22 Cécile Gachet
Université Côte d'Azur
The cone conjecture for certain Calabi-Yau manifolds obtained as fiber products The cone conjecture is a long-lasting conjecture in birational geometry, and can be viewed as a conjectural counterpart to the cone theorem, for K-trivial varieties. It predicts for any K-trivial variety X the existence of a rational polyhedral fundamental domain for the action of the group Aut(X) on the cone Nefe(X). If the conjecture is true for a certain X, it has important consequences, such as the finiteness of real structures on X, or the fact that the group Aut(X)/Aut0(X) is finitely presented. In this talk, I present a construction inspired by the work of Schoen, Namikawa and Grassi-Morrison, of Calabi-Yau manifolds obtained as fiber products over the projective line, and prove the cone conjecture for these Calabi-Yau manifolds. This is joint work with Hsueh-Yung Lin and Long Wang.
November 29 Ahmed Ashem Abouelsaad
University of Basel
Galois Points and Cremona Transformations We study the Galois points for any irreducible plane curve C subset from the two dimensional projective space of degree d (d greater than or equal 1) we prove that a Galois group of order at most 3 always can be extended to a group of de Jonquières Cremona transformation. We also show that there are Galois groups that can be extended to Cremona transformations but not de Jonquières maps, and there are Galois groups that cannot be extended to Cremona transformations.
December 6 Irène Meunier
University of Basel
A study of dynamical degrees of polynomial automorphisms of the affine space via their action on the space of valuations We are interested in computing the dynamical degree of polynomial automorphisms of ℂn. It seems that interesting data about the dynamics can be extracted from the action of automorphisms over the space of valuations. We will start by defining the space of valuations. More precisely, we will focus on a subspace of it with good geometric properties. One can see this space as generated by the action of tame polynomial automorphisms over the space of monomial valuations. Then, for a fixed polynomial endomorphism of ℂn, we will define a degree function as a generalisation of the standard degree, defined over the whole space of valuations. We will then focus on the dimension three and evoke our research. We will try to explain how good properties of our degree, in connection with these geometrical properties of our space could lead to get an algorithmic way to compute the dynamical degree of any (tame) polynomial endomorphism.
December 20 Adrien Dubouloz
Université de Bourgogne (CNRS)
Algebraic families of real forms (Joint work in progress with Anna Bot) Real forms of a given real algebraic variety X, that is, real algebraic varieties whose complexifications are isomorphic to that of X, are classified after Borel-Serre in terms of the first Galois cohomology set of the group of automorphisms of the complexification of X. A result of T. Labinet asserting in particular that every complex algebraic group scheme locally of finite type has finite first Galois cohomology set implies that every real projective variety has at most countably many real forms. In contrast, some examples of affine and strictly quasi-projective smooth real varieties -with large non-algebraic automorphism groups- having uncountably many real forms have been recently constructed. These examples motivate the study of real forms from a moduli viewpoint, a typical question being whether forms of a given real variety can be organized in families depending algebraically on the real points of a suitable algebraic variety. In the talk I will propose and illustrate on examples different possible notions of "algebraic families of real forms" of a given real algebraic variety. Time permitting, I will also outline the steps of a construction for every real algebraic variety X of a natural complete family of real forms of X parametrized by a "space" whose real points are in natural bijection with the set of Galois 1-cocycles with values in the automorphism group of the complexification of X.