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SUMMARY:k-gonal loci in Severi varieties of curves on K3 surfaces and rati
 onal curves on hyperkahler manifolds - Andreas Knutsen (Bergen)
DTSTART:20130501T131500Z
DTEND:20130501T141500Z
UID:TALK43933@talks.cam.ac.uk
CONTACT:Dr. J Ross
DESCRIPTION:This is a report on recent joint work with C. Ciliberto. Let $
 (S\,H)$ be a general primitively polarized complex K3 surface of genus $p$
  and consider the Severi varieties $V_{|H|\,\\delta}$ parametrizing $\\del
 ta$- nodal curves in the linear system $|H|$\, for $0 \\leq \\delta \\leq 
 p$.\n\nIt is well-known that these are smooth and nonempty of dimension $p
 -\\delta$. We consider the subloci V(k\,|H|\,\\delta) of  curves whose nor
 malizations possess a g1k. We give necessary and sufficient\nconditions d
 epending on $p$\, $\\delta$ and $k$ for these loci to be nonempty\, and pr
 ove that\, when nonempty\, there is always an irreducible component of the
 \nexpected dimension. In contrast to the case of smooth curves\, the Sever
 i varieties thus contain proper subloci of gonalities lower than the maxim
 al gonality\ngiven by Brill-Noether theory.\n\nI will also discuss applica
 tions to the study of the Mori cone of rational curves in the punctual Hil
 bert scheme Hilb(k\,S)$\, since the curves in V(k\,|H|\,\\delta) naturally
  induce rational curves in\n\\Hilb(k\,S)$.
LOCATION:MR 13\, CMS
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