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SUMMARY:Leonid Gorodetskii (MIT)
DTSTART:20260402T210000Z
DTEND:20260402T223000Z
DTSTAMP:20260422T220549Z
UID:STAGE/154
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/STAGE/154/">
 Universal torsors</a>\nby Leonid Gorodetskii (MIT) as part of STAGE\n\nLec
 ture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nIn this t
 alk\, we focus on torsors under algebraic tori. For an algebraic variety $
 X$ with discrete $\\operatorname{Pic}(X_{\\bar k})$ (for instance\, when $
 X_{\\bar k}$ is rationally connected)\, the dual group $T = \\operatorname
 {Hom}(\\operatorname{Pic}(X_{\\bar k})\, \\mathbb{G}_m)$ is a natural alge
 braic torus associated to $X$. Universal torsors are a class of torsors ov
 er $X$ under $T$ which\, on the one hand\, can capture the set of all rati
 onal points on $X$ and\, on the other\, often admit an explicit descriptio
 n. After developing the theory\, we will see how universal torsors can be 
 used to prove the existence of rational points.\n\nReferences:\n<br>\n- <a
  href="https://www.math.univ-paris13.fr/~wittenberg/slc.pdf">Wittenberg</a
 >\, Rational points and zero-cycles on rationally connected varieties over
  number fields\, Section 3.3. \n<br>\n- <a href="https://doi.org/10.1017/C
 BO9780511549588">Skorobogatov</a>\, Torsors and Rational Points\, Section 
 2.3\, p.25 and the references there.\n
LOCATION:https://researchseminars.org/talk/STAGE/154/
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