Linear equivalence and Hilbert function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:35:11Z http://mathoverflow.net/feeds/question/94110 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94110/linear-equivalence-and-hilbert-function Linear equivalence and Hilbert function Naga Venkata 2012-04-15T12:39:41Z 2013-01-09T16:22:00Z <p>Let \$X \subset \mathbb{P}^3\$ be a smooth degree \$d\$ surface containing two irreducible curves \$C_1, C_2\$ linearly equivalent to each other. If we assume that \$X\$ is general (among all degree \$d\$ smooth surfaces in \$\mathbb{P}^3\$) then is it true that \$I_d(C_1)=I_d(C_2)\$?</p> http://mathoverflow.net/questions/94110/linear-equivalence-and-hilbert-function/117260#117260 Answer by Dmitri for Linear equivalence and Hilbert function Dmitri 2012-12-26T15:27:16Z 2012-12-26T15:27:16Z <p>Let me show that the answer to this question is positive for \$d>3\$. Indeed, for a general surface \$X\$ of degree \$d>3\$ its Picard group is \$\mathbb Z\$ and is generated by \$O(1)\$. It follows that both curves \$C_1\$ and \$C_2\$ are complete intersections, and so they have the same Hibert function (see for example Section 13 pages 172-173 in book of Harris "first course in algebraic geometry"). Hence the statement is proved. </p>