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.
Date  Speaker  Title  Abstract 
September 20 
Marc Abboud
Rennes 
Dynamical degrees of endomorphisms of complex affine surfaces are quadratic integers

Let X_{0} be a complex affine surface and let f be a dominant endomorphism of X_{0}. The dynamical degree λ_{1}
of f is defined as follows. Let X be a projective surface with an embedding ι: X_{0} ↪ X such that
ι(X_{0}) is an open dense subset of X. Take any ample divisor H on X and define λ_{1} := lim_{n
→ ∞} ( ( f^{n} )^{*} H \cdot H )^{1/n}. This does not depends on H nor on the
projective surface X; one has λ_{1}^{2} ≥ e where e is the topological degree of f. In
2007, Favre and Jonsson showed when X_{0} 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 X_{0}. The main result is that when λ_{1}^{2}
> 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 X_{0} 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 SL_{2}(k) on affine kdomains via fundamental pairs

Let k be a field of characteristic zero. In their 2012 paper, Arzhantsev and Liendo write:
" A regular SL_{2}action on an kaffine variety X is uniquely defined by an
𝖘𝔩_{2}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 kdomain B as a pair (D,U) which satisfies
 [D,[D,U]]=2D and [U,[D,U]]=2U, and
 B=⊕_{d∈ℤ}B_{d} is a ℤgrading where B_{d}=ker ([D,U]dI).
Let A⊂ B be the kernel of D. Classical techniques show that A is affine. The Structure Theorem describes A as an ℕgraded ring, as well as the degree modules and image ideals of D.
The Structure Theorem is used to classify normal affine SL_{2}(k)surfaces with trivial units, generalizing the classification given by Gizatullin and Popov for complex SL_{2}(ℂ )surfaces.
This classification is used to describe threedimensional UFDs which admit a certain kind of SL_{2}(k) action.
From this, we show that any SL_{2}(k)action on 𝔸_{k}^{3} 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 SL_{2}(k)actions on 𝔸_{k}^{4}, 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 𝔸_{k}^{1}\cong𝔸_{k}^{4}. If X admits a nontrivial action of SL_{2}(k), then X is isomorphic to 𝔸_{k}^{3}.
We close with several open questions about SL_{2}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

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November 8 
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November 15 
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November 22 
Cécile Gachet
Université Côte d'Azur 
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November 29 
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December 6 
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December 6 
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December 13 
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December 20 
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