Is there some short formula for the "defect" of Hilbert function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:45:43Z http://mathoverflow.net/feeds/question/64404 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64404/is-there-some-short-formula-for-the-defect-of-hilbert-function Is there some short formula for the "defect" of Hilbert function Dmitry Kerner 2011-05-09T15:58:44Z 2011-05-09T16:06:40Z <p>Let $X\subset\Bbb P^n_{\Bbb C}$ be a connected, Cohen Macaulay sub-scheme. (Possibly singular, reducible or non-reduced.) For $k\gg0$ the numbers $h^0(\mathcal{O}_X(k))$ depend polynomially on $k$. Denote this polynomial by $P(k):\stackrel{k\gg0}{=}h^0(\mathcal{O}_X(k))$ .</p> <p>What is the geometric meaning of the "total defect" $\sum_{k\ge0}\Big(h^0(\mathcal{O}_X(k))-P(k)\Big)$?</p> <p>e.g. if $X\subset\Bbb P^n_{\Bbb C}$ is a hypersurface of degree $d$ then this total defect is ${d\choose{n}}$, which can be interpreted in various ways. (Arithmetic genus, Euler characteristic etc.) Similarly for complete intersections. Is there some general statement? A reference?</p> <p>I guess this defect is constant under flat deformations of $X$ in $\Bbb P^n$, and I'm interested in the smoothable case. So probably can assume $X$ smooth? (Does this help?)</p> <p>upd. (I guess it helps.) In my case the embedding of $X$ is projectively normal. (Alternatively, $\mathcal{O}_X(k)$ is normally generated.)</p>