What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T08:45:18Z http://mathoverflow.net/feeds/question/12068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12068/what-is-the-euler-characteristic-of-a-hilbert-scheme-of-points-of-a-singular-alge What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? James O 2010-01-17T03:13:26Z 2010-09-23T16:17:55Z <p>Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula computing the Euler characteristic of all these n-symmetrical products:</p> <p>$$\sum_{d \geq 0} \ \chi \left(X^{[d]} \right)q^d \ \ = \ \ (1-q)^{- \chi(X)}$$ </p> <p>It is known that $S^nX \cong X^{[n]}$, the Hilbert scheme of 0-subschemes of length n over $X$. Hence, the previous formula also computes the Euler characteristic of these spaces. </p> <p>What about for singular surfaces? More precisely, if $X$ is a singular complex algebraic curve, do you know how to compute the Euler characteristic of its n-symmetrical powers $S^nX$? More importantly: what is the Euler characteristic of $X^{[n]}$, the Hilbert scheme of 0-schemes of length n over $X$?</p> <p>I guess it is too much to hope for a formula as neat as the one given for the smooth case. Examples, formulas for a few cases or general behaviour (e.g. if for large n, $\chi\left(X^{[n]}\right) = 0)$ are all very welcome!</p> http://mathoverflow.net/questions/12068/what-is-the-euler-characteristic-of-a-hilbert-scheme-of-points-of-a-singular-alge/12069#12069 Answer by Tim Perutz for What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? Tim Perutz 2010-01-17T03:57:13Z 2010-01-17T04:49:06Z <p>(<i>Edited to reflect adjusted question.</i>) The topology of symmetric products of a non-singular curve was worked out by Macdonald, Topology 1 (1962), 319-343. The starting point is the observation (which is quite general, hence applies to singular curves) that the rational cohomology of the symmetric product is the invariant part of the cohomology of the Cartesian product, from which one can work out the Euler characteristic.</p> <p>The Euler characteristic of the Hilbert scheme of $n$ points on a curve is a rather subtler problem, since these spaces feel the nature of the singularities. In the simplest case, a curve with one node, I wrote down how to describe the Hilbert scheme as a variety with normal crossings <a href="http://arxiv.org/abs/math/0606061" rel="nofollow">here</a> (see Section 3.2 and the appendix). I also described its cohomology. From that you can write down the Euler characteristic.</p> http://mathoverflow.net/questions/12068/what-is-the-euler-characteristic-of-a-hilbert-scheme-of-points-of-a-singular-alge/12071#12071 Answer by VA for What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? VA 2010-01-17T04:07:20Z 2010-01-17T04:13:23Z <p>The same formula holds for $\chi(X^{[n]})$ of any topological space for which $\chi(X)$ is defined and behaves in the expected way for unions, Cartesian products, and quotients by a finite free action. This includes the category of all algebraic varieties over $\mathbb C$.</p> <p>For example, $$\chi(X^{[2]})= \frac{\chi(X\times X)- \chi(\operatorname{diag} X)}{2} + \chi(X) = \frac{ \chi(X)^2 + \chi(X) }{2},$$ etc. No matter whether $X$ is a smooth curve or a singular 3-fold.</p> http://mathoverflow.net/questions/12068/what-is-the-euler-characteristic-of-a-hilbert-scheme-of-points-of-a-singular-alge/13232#13232 Answer by Va P for What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? Va P 2010-01-28T06:34:20Z 2010-01-28T06:34:20Z <p>It is the partitions of n p(n) as is shown in this paper by De Cataldo <a href="http://www.springerlink.com/content/gab4epw8tmt4hk4a/" rel="nofollow">http://www.springerlink.com/content/gab4epw8tmt4hk4a/</a></p> http://mathoverflow.net/questions/12068/what-is-the-euler-characteristic-of-a-hilbert-scheme-of-points-of-a-singular-alge/18540#18540 Answer by Vivek Shende for What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? Vivek Shende 2010-03-18T02:29:25Z 2010-09-23T16:17:55Z <p>For singular plane curves, there is a conjectural formula (due to Alexei Oblomkov and myself) in terms of the HOMFLY polynomial of the links of the singularities. For curves whose singularities are torus knots, i.e. like x^a = y^b for a,b relatively prime, and for a few more singularities, the conjecture has been established. See <a href="http://arxiv.org/abs/1003.1568" rel="nofollow">this preprint</a>. </p> <p><strong>Edit:</strong> More recently I have given <a href="http://arxiv.org/abs/1009.0914" rel="nofollow">a different characterization</a> of these numbers in terms of multiplicities of certain strata in the versal deformation of the singular curve.</p>