## 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.
DateSpeakerTitleAbstract
September 20 Marc Abboud
Rennes
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
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October 18 Gebhard Martin
Bonn
Automorphisms of del Pezzo surfaces in positive characteristic tba
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November 22 Cécile Gachet
Université Côte d'Azur
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