COOKIES: By using this website you agree that we can place Google Analytics Cookies on your device for performance monitoring. |

University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Refined curve counting on algebraic surfaces

## Refined curve counting on algebraic surfacesAdd to your list(s) Download to your calendar using vCal - Goettsche, L (ICTP)
- Monday 27 June 2011, 16:30-17:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact Mustapha Amrani. Moduli Spaces Let $L$ be ample line bundle on an an algebraic surface $X$. If $L$ is sufficiently ample wrt $d$, the number $t_d(L)$ of $d$-nodal curves 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 define the numbers $t_d(L)$ as BPS invariants and prove a conjecture of mine about their generating function (proved by Tzeng using different methods). We use the generating function of the $i_y$-genera of these relative 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 invariants to the Donaldson-Thomas invariants considered by Maulik-Pandharipande-Thomas. In the case of toric surfaces we find that the refined invariants 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 recursion, which specializes (at $y=1,-1$) to the corresponding recursion for complex curves and the Welschinger invariants. This is in part joint work with Vivek Shende. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
- Featured lists
- INI info aggregator
- Isaac Newton Institute Seminar Series
- School of Physical Sciences
- Seminar Room 1, Newton Institute
Note that ex-directory lists are not shown. |
## Other listsCambridge Rare Earths Society Neuroscience Judge Business Club Financial Economcs Series## Other talksSitting Pretty: Local Representation in China's Party Leadership Bodies Design a Resillient Manufacturing System in the Presence of Change Hands-on Generative Programming: Write an Interpreter, Build a Compiler. Aristotelian Logic for Engineers Quaternary Discussion Group seminar A tale of sleepless flies and ninna nanna. How Drosophila changes what we know about sleep. |