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

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