Título: Cox rings of K3 surfaces

Palestrante: Michela Artebani (Universidad de Concepción)
Data: 17/11/2020
Horário: 15:00 GMT
Local: Transmissão online.

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Resumo: Given a normal complex projective variety X with finitely generated divisor class group, its Cox ring R(X) is the Cl(X)-graded algebra whose homogeneous pieces are Riemann-Roch spaces of divisors of X. This object is particularly interesting when it is finitely generated, since in such case X can be obtained as a GIT quotient of an open subset of Spec R(X) by the action of a quasi-torus¹. Finding a presentation or even a minimal generating set for R(X) is in general a difficult problem, already in the case of surfaces. In this talk, after an introduction to the subject, we will concentrate on complex projective K3 surfaces, which are known to have finitely generated Cox ring exactly when their automorphism group is finite². We show that the Cox ring can be generated by homogeneous elements whose degrees are either classes of (-2)-curves, sums of at most three elements in the Hilbert basis of the nef cone, or classes of divisors of the form 2(E+E'), where E,E' are elliptic curves with E.E'=2. As an application, we compute Cox rings of Mori dream K3 surfaces of Picard number 3 and 4.
This is joint work with C. Correa Deisler, A. Laface and X. Roulleau^3,4

References:
^[1] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015.
^[2] M. Artebani, J. Hausen, and A. Laface, On Cox rings of K3 surfaces, Compos. Math. 146 (2010), no. 4, 964–998. arXiv:0901.0369
^[3] M. Artebani, C. Correa Deisler, and A. Laface, Cox rings of K3 surfaces of Picard number three, J. Algebra 565C (2021), 598–626. arXiv:1909.01267
^[4] M. Artebani, C. Correa Deisler, and X. Roulleau, Mori dream K3 surfaces of Picard number four: projective models and Cox rings. arXiv:2011.00475

Host: Antonella Grassi
Zoom ID: 991 849 3831
Password: Lomonosov

Título: Mixed Effects State Space Models for Longitudinal Data with Heavy Tails

Autor: Lina Lucia Hernandez Velasco
Data: 09/11/20
Horário: 10:00h
Local: Transmissão online

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Banca Examinadora:
Aarlos Antonio Abanto-Valle - Presidente (IM - UFRJ)
Fernando Antonio da Silva Moura (IM - UFRJ)
Kelly Cristina Mota Gonçalves (IM - UFRJ)
Victor Hugo Lachos Dávila (UNICAMP)
Luis Mauricio Castro Cepero (PUC-Chile)
Marina Silva Paez (IM - UFRJ)

Título: Rationally connected rational double covers of primitive Fano varieties

Palestrante: Aleksandr Pukhlikov (University of Liverpool)
Data: 10/11/2020
Horário: 15:00 GMT (12h BRT)
Local: Transmissão online.

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Resumo: We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties.

Host: Yuri Prokhorov
Zoom ID: 991 849 3831
Password: Lomonosov

No dia 27 de outubro de 2020, recebemos a visita da reitora da UFRJ, Professora Denise Carvalho, e do Prefeito Marcus Maldonado, junto à Direção do IM nas obras do prédio do Instituto de Matemática da UFRJ.