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SUMMARY:Roberto Villaflor Loyola (Universidad Técnica Federico Santa Mar
 ía)
DTSTART:20260403T124000Z
DTEND:20260403T134000Z
DTSTAMP:20260423T021258Z
UID:OBAGS/83
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OBAGS/83/">O
 n the linear cycles conjecture</a>\nby Roberto Villaflor Loyola (Universid
 ad Técnica Federico Santa María) as part of ODTU-Bilkent Algebraic Geome
 try Seminars\n\n\nAbstract\nThe classical Noether-Lefschetz theorem claims
  that a very general degree $d>3$ surface in $\\mathbb{P}^3$ has Picard nu
 mber one. The locus of surfaces with higher Picard rank is known as the No
 ether-Lefschetz locus\, which is known to have a countable number of irred
 ucible components. For $d>4$\, it is classical result due independently to
  Green and Voisin\, that the unique component of highest codimension corre
 sponds to the locus of surfaces which contain lines. \n\nThe natural gener
 alization of this question to higher dimensional hypersurfaces of the proj
 ective space is known as the "<em>linear cycles conjecture</em>"\, and rem
 ains open even for fourfolds. For surfaces\, the proof is based in the fac
 t that locally (analytically) one can parametrize each component by a Hodg
 e locus\, and then use the Infinitesimal Variation of Hodge Structure to c
 ompute (and bound) the dimension of its Zariski tangent space. \n\nA natur
 al stronger version of the linear cycles conjecture is that the Hodge loci
  with maximal tangent space are those corresponding to linear cycles. \n\n
 In this talk I will report on recent results disproving this conjecture fo
 r all degrees and dimensions. \n\nThis is a joint work with Jorge Duque Fr
 anco.\n
LOCATION:https://researchseminars.org/talk/OBAGS/83/
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