On one of the ends of MMP: Markovian planes
Friday, 20.10.23, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
About a year ago, I gave a talk at this seminar on degenerations of surfaces with Wahl singularities. I tried by then to explain the explicit birational picture and some connections with exceptional collections of vector bundles. We see in this Mori theory divisorial contractions and flips controlled by Hirzebruch-Jung continued fractions (a summary can be found here https://arxiv.org/abs/1311.4844). As final products of this MMP, we arrive at either nef canonical class, smooth deformations of ruled surfaces, and degenerations of the projective plane (compare with the classical MMP for nonsingular projective surfaces). In this talk, I would like to explain these "Markovian planes". The name comes from the classification of such degenerations , due to Hacking and Prokorov 2010 (after Badescu and Manetti), as partial smoothings of P(a^2,b^2,c^2) where (a,b,c) satisfies the Markov equation x^2+y^2+z^2=3xyz. It turns out that there is a beautiful birational picture behind them, which in particular gives new insights to Markov's uniqueness conjecture. This is a joint work in progress together with Juan Pablo Zúñiga (Ph.D. student at UC Chile). \n\n
Arithmetic aspects of quantum cohomology
Friday, 15.12.23, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Shintanis (wenig bekannte) Vermutung zum 12. Hilbertschen Problem besagt, dass abelsche Erweiterungen von \nZahlkörpern mit Hilfe spezieller Werte von verallgemeinerten Gammafunktionen erzeugt werden können.\n\nDiese Vermutung ist seit gut 50 Jahren ungelöst und weitestgehend unverstanden. Unser Vortrag wird daran leider nichts ändern.\n\nEine Hauptschwierigkeit der Shintanischen Vermutung liegt darin, dass (verallgemeinerte) Gammafunktionen sehr schwer handhabbar sind.\n\nIn unserem (elementar gehaltenen) Vortrag wollen wir erklären, wie der Formalismus der Quantumkohomologie neue Einsichten in das Wesen der Gammafunktionen\nliefern könnte.
Neues zur Additivität der Kodaira-Dimension
Friday, 9.2.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Cohomology of punctual Hilbert schemes of smooth projective surfaces
Monday, 4.3.24, 10:15-11:15, Raum 404, Ernst-Zermelo-Str. 1
The punctual Hilbert scheme X^[n] of a projective scheme X over a field k is a scheme parameterising the closed subschemes of X of length n; loosely speaking, those that have n points when counted with multiplicity. It turns out that if X is smooth, X^[n] is necessarily smooth as well if and only if dim X < 3, making the case where X is a smooth projective surface of particular interest. Remarkable work by L. Göttsche demonstrated that if k is C or the algebraic closure of a finite field, the Betti numbers and the Euler characteristic of X^[n] can be written in terms of explicit generating functions related to modular forms. This talk will review some properties of punctual Hilbert schemes in general, study the special case of projective surfaces and discuss their cohomology.