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UID:news1983@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20260211T132241
DTSTART;TZID=Europe/Zurich:20260218T150000
SUMMARY:Seminar Algebra and Geometry: Antonio Laface (Universidad de Concep
 ción)
DESCRIPTION:In 1959\, Nagata produced a counterexample to Hilbert’s 14th 
 problem via a linear action of a non-reductive group. The resulting non-fi
 nitely generated invariant algebra can be interpreted as the Cox ring of a
  blow-up of P^2 at points\, and in this framework he formulated his conjec
 ture on plane curves. More recently\, in joint work with Castravet\, Tevel
 ev and Ugaglia we show that if the blow-up of certain toric surfaces at a 
 general point has a non-polyhedral pseudoeffective cone\, then the pseudoe
 ffective cone of M̄_{0\,n} is not polyhedral for n ≥ 10. These results 
 motivate a systematic study of Cox rings of blow-ups of toric varieties at
  a general point.In this talk I focus on minimal toric surfaces\, namely t
 hose without curves of negative self-intersection. Such surfaces are cycli
 c quotients of either P^2 or P^1 x P^1. For quotients of P^2\, the blow-up
  at a general point may fail to be a Mori Dream Space\, for instance when 
 the semiample cone is not closed\, or when the effective cone is not close
 d. In contrast\, for quotients of P^1 x P^1\, we prove that the blow-up at
  a general point is always a Mori Dream Space.This is joint work with Luca
  Ugaglia.
X-ALT-DESC:<p>In 1959\, Nagata produced a counterexample to Hilbert’s 14t
 h problem via a linear action of a non-reductive group. The resulting non-
 finitely generated invariant algebra can be interpreted as the Cox ring of
  a blow-up of P^2 at points\, and in this framework he formulated his conj
 ecture on plane curves. More recently\, in joint work with Castravet\, Tev
 elev and Ugaglia we show that if the blow-up of certain toric surfaces at 
 a general point has a non-polyhedral pseudoeffective cone\, then the pseud
 oeffective cone of M̄_{0\,n} is not polyhedral for n ≥ 10. These result
 s motivate a systematic study of Cox rings of blow-ups of toric varieties 
 at a general point.<br /><br />In this talk I focus on minimal toric surfa
 ces\, namely those without curves of negative self-intersection. Such surf
 aces are cyclic quotients of either P^2 or P^1 x P^1. For quotients of P^2
 \, the blow-up at a general point may fail to be a Mori Dream Space\, for 
 instance when the semiample cone is not closed\, or when the effective con
 e is not closed. In contrast\, for quotients of P^1 x P^1\, we prove that 
 the blow-up at a general point is always a Mori Dream Space.<br /><br />Th
 is is joint work with Luca Ugaglia.</p>
DTEND;TZID=Europe/Zurich:20260218T160000
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