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SUMMARY:Refined curve counting on algebraic surfaces - Goettsche\, L (ICTP
 )
DTSTART:20110627T153000Z
DTEND:20110627T163000Z
UID:TALK31894@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Let $L$ be ample line bundle on an an algebraic surface $X$. I
 f $L$ is sufficiently ample wrt $d$\, the number $t_d(L)$ of $d$-nodal cur
 ves in a general $d$-dimensional sub linear system of |L| will be finite. 
 Kool-Shende-Thomas use the generating function of the Euler numbers of the
  relative Hilbert schemes of points of the universal curve over $|L|$ to d
 efine the numbers $t_d(L)$ as BPS invariants and prove a conjecture of min
 e about their generating function (proved by Tzeng using different methods
 ). \n\nWe use the generating function of the $i_y$-genera of these relati
 ve Hilbert schemes to define and study refined curve counting invariants\,
  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 invar
 iants to the Donaldson-Thomas invariants considered by Maulik-Pandharipand
 e-Thomas. \n\nIn the case of toric surfaces we find that the refined invar
 iants interpolate between the Gromow-Witten invariants (at $y=1$) and the 
 Welschinger invariants at $y=-1$. We also find that refined invariants of 
 toric surfaces can be defined and computed by a Caporaso-Harris type recur
 sion\, which specializes (at $y=1\,-1$) to the corresponding recursion for
  complex curves and the Welschinger invariants. \n\nThis is in part joint 
 work with Vivek Shende.\n
LOCATION:Seminar Room 1\, Newton Institute
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