When are Hilbert schemes smooth? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T16:57:25Z http://mathoverflow.net/feeds/question/244 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/244/when-are-hilbert-schemes-smooth When are Hilbert schemes smooth? Kevin Lin 2009-10-09T22:43:45Z 2012-06-28T20:10:32Z <p>I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?</p> http://mathoverflow.net/questions/244/when-are-hilbert-schemes-smooth/247#247 Answer by Ben Webster for When are Hilbert schemes smooth? Ben Webster 2009-10-10T01:35:18Z 2009-10-11T19:14:41Z <p>A very well-known condition is that the Hilbert scheme of a smooth surface is smooth. As David pointed out below, the Hilbert scheme of a smooth curve is smooth and equal to the symmetric product (since k[t] has only one finite dimension quotient of each dimension). </p> <p>I don't know of any other examples, but one of the versions of <a href="http://arxiv.org/abs/math/0411469" rel="nofollow">Murphy's Law in algebraic geometry</a> is roughly "if you don't have a good reason for a Hilbert scheme to not be horrible, it will be as horrible as you can possibly imagine."</p> http://mathoverflow.net/questions/244/when-are-hilbert-schemes-smooth/251#251 Answer by David Speyer for When are Hilbert schemes smooth? David Speyer 2009-10-10T03:48:37Z 2009-10-10T03:48:37Z <p>The Hilbert scheme of n points on a 3-fold is not smooth for n sufficiently large, but the exact value of sufficiently large is unknown. See the chapter on Hilbert schemes in "Combinatorial Commutative Algebra", by Miller and Sturmfels.</p> http://mathoverflow.net/questions/244/when-are-hilbert-schemes-smooth/299#299 Answer by Daniel Erman for When are Hilbert schemes smooth? Daniel Erman 2009-10-11T18:39:24Z 2009-10-11T18:39:24Z <p>I don't know of many global conditions for a Hilbert scheme to be smooth/singular. Ben's answer probably gives the most interesting example of a smooth Hilbert scheme, namely the Hilbert scheme of n points on a smooth surface.</p> <p>Here are two more examples of smooth Hilbert schemes.</p> <p>1) The Hilbert scheme of hypersurfaces of degree d in PP^n. Such hypersurfaces are parametrized by homogeneous degree d polynomials in n+1 variables, and hence this Hilbert scheme is a projective space of dimension n+d choose d.</p> <p>2) The Hilbert scheme of linear subpsace of dimension d of PP^n. This is just the Grassmanian Gr(d+1,n+1).</p> http://mathoverflow.net/questions/244/when-are-hilbert-schemes-smooth/899#899 Answer by mdeland for When are Hilbert schemes smooth? mdeland 2009-10-17T17:27:14Z 2012-06-28T20:10:32Z <p>Here is yet another example of a smooth Hilbert scheme. Let \$X\$ be a smooth degree 3 hypersurface in projective space of dimension \$n \geq 3\$ (say, over an algebraically closed field), and let \$H\$ be the Hilbert scheme of lines on \$X\$ (i.e., corresponding to Hilbert polynomial \$t + 1\$). </p> <p>The tangent space to \$H\$ at a point \$[L]\$ (corresponding to a line \$L\$ in \$X\$) is \$H^0(L, N)\$ where \$N\$ is the normal bundle of \$L\$ in \$X\$. The rank of \$N\$ is \$n - 2\$ and the degree of \$N\$ is \$2n - 6\$ (you can see this by looking at the standard tangent bundle and normal bundle sequences). Every vector bundle on \$L = \mathbb{P}^1\$ splits into the direct sum of line bundles. Then the degree of each rank 1 summand of \$N\$ is at most 1 (\$N\$ injects into the normal bundle of \$L\$ in \$\mathbb P^n\$) and then you can show that no piece can have degree less than \$-1\$. This allows us to conclude that \$H^1(L, N) = 0\$. This means that \$H\$ is smooth at the point \$[L]\$ (see for example Kollár's book <em>Rational Curves on Algebraic Varieties</em>, Chapter 1, where he explains the infinitesimal behavior of the Hilbert scheme). Since this is true for any line \$L\$ in \$X\$, the Hilbert scheme is smooth. </p> <p>The same argument works for lines on a smooth Quadric. In the same book, Kollár proves that for a general degree \$d\$ hypersurface \$X\$ in \$\mathbb P^n\$, the Hilbert scheme of lines on \$X\$ is smooth. </p>