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SUMMARY:Michela Artebani (Universidad de Concepcion)
DTSTART:20201117T150000Z
DTEND:20201117T160000Z
DTSTAMP:20260423T021444Z
UID:ZAG/69
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ZAG/69/">Cox
  rings of K3 surfaces</a>\nby Michela Artebani (Universidad de Concepcion)
  as part of ZAG (Zoom Algebraic Geometry) seminar\n\n\nAbstract\nGiven a n
 ormal complex projective variety X with finitely generated divisor class g
 roup\, its Cox ring R(X) is the Cl(X)-graded algebra whose homogeneous pie
 ces 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 ob
 tained as a GIT quotient of an open subset of Spec R(X) by the action of a
  quasi-torus [1]. Finding a presentation or even a minimal generating set 
 for R(X) is in general a difficult problem\, already in the case of surfac
 es. In this talk\, after an introduction to the subject\, we will concentr
 ate on complex projective K3 surfaces\, which are known to have finitely g
 enerated Cox ring exactly when their automorphism group is finite [2]. We 
 show that the Cox ring can be generated by homogeneous elements whose degr
 ees 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. Thi
 s is joint work with C. Correa Deisler\, A. Laface and X. Roulleau [3\,4].
 \n\nReferences.\n[1] I. Arzhantsev\, U. Derenthal\, J. Hausen\, and A. Laf
 ace\, Cox rings\, Cambridge Studies in Advanced Mathematics\, vol. 144\, C
 ambridge University Press\, Cambridge\, 2015.\n[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\n[3] M. Artebani\, C. Correa Deisler\
 , and A. Laface\, Cox rings of K3 surfaces of Picard number three\, J. Alg
 ebra 565C (2021)\, 598–626. arXiv:1909.01267\n[4] M. Artebani\, C. Corre
 a Deisler\, and X. Roulleau\, Mori dream K3 surfaces of Picard number four
 : projective models and Cox rings. arXiv:2011.00475.\n
LOCATION:https://researchseminars.org/talk/ZAG/69/
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