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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Refined curve counting on algebraic surfaces - Goe
ttsche\, L (ICTP)
DTSTART;TZID=Europe/London:20110627T163000
DTEND;TZID=Europe/London:20110627T173000
UID:TALK31894AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/31894
DESCRIPTION:Let $L$ be ample line bundle on an an algebraic su
rface $X$. If $L$ is sufficiently ample wrt $d$\,
the number $t_d(L)$ of $d$-nodal curves in a gener
al $d$-dimensional sub linear system of |L| will b
e finite. Kool-Shende-Thomas use the generating fu
nction of the Euler numbers of the relative Hilber
t schemes of points of the universal curve over $|
L|$ to define the numbers $t_d(L)$ as BPS invarian
ts and prove a conjecture of mine about their gene
rating function (proved by Tzeng using different m
ethods). \n\nWe use the generating function of the
$i_y$-genera of these relative Hilbert schemes t
o define and study refined curve counting invarian
ts\, which instead of numbers are now polynomials
in $y$\, specializing to the numbers of curves for
$y=1$. If $X$ is a K3 surface we relate these inv
ariants to the Donaldson-Thomas invariants conside
red by Maulik-Pandharipande-Thomas. \n\nIn the cas
e of toric surfaces we find that the refined invar
iants interpolate between the Gromow-Witten invari
ants (at $y=1$) and the Welschinger invariants at
$y=-1$. We also find that refined invariants of to
ric surfaces can be defined and computed by a Capo
raso-Harris type recursion\, which specializes (at
$y=1\,-1$) to the corresponding recursion for com
plex curves and the Welschinger invariants. \n\nTh
is is in part joint work with Vivek Shende.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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