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SUMMARY:Quadratic Welschinger invariants - Marc Levine (Universität Duisb
 urg-Essen)
DTSTART:20180814T080000Z
DTEND:20180814T090000Z
UID:TALK108679@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:This is report on part of a program to give refinements of num
 erical invariants arising in enumerative geometry to invariants living in 
 the Grothendieck-Witt ring over the base-field. Here we define an invarian
 t in the Grothendieck-Witt ring for ``counting&#39\;&#39\; rational curves
 . More precisely\, for a del Pezzo surface S over a field k and a  positiv
 e degree curve class $D$ (with respect to the anti-canonical class $-K_S$)
 \, we define a class in the Grothendiek-Witt ring of k\, whose rank gives 
 the number of rational curves in the class D  containing a given collectio
 n of distinct closed points $\\mathfrak{p}=\\sum_ip_i$ of total degree $-D
 \\cdot K_S-1$. This recovers Welschinger&#39\;s invariants in case $k=\\ma
 thbb{R}$ by applying the signature map. The main result is that this quadr
 atic invariant depends only on the $\\mathbb{A}^1$-connected component con
 taining $\\mathfrak{p}$ in $Sym^{3d-1}(S)^0(k)$\, where  $Sym^{3d-1}(S)^0$
  is the open subscheme of $Sym^{3d-1}(S)$ parametrizing geometrically redu
 ced 0-cycles.
LOCATION:Seminar Room 1\, Newton Institute
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