Genus of non-complete intersections - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:01:54Z http://mathoverflow.net/feeds/question/114483 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114483/genus-of-non-complete-intersections Genus of non-complete intersections LMN 2012-11-26T03:58:10Z 2012-11-26T03:58:10Z <p>Suppose $X\subset \mathbb{P}_k^N$ a nonsingular curve is a complete intersection of hypersurfaces $F_1, \cdots, F_{N-1}$ (of degrees $d_1, \cdots, d_{N-1}$ resp). Then, we know that the canonical divisor on $X$ is $\mathcal{O}_X(d_1 + \cdots + d_{N-1} - n - 1)$. Hence, intersection theory on projective space gives a formula for the genus of $X$ entirely in terms of the various $d_i$. Specifically, $$2g - 2 = d_1\cdots d_{N-1} (d_1 + \cdots + d_{N-1} - N - 1)$$</p> <p>$\textbf{Question:}$ Suppose $X$ is a nonsingular curve in $\mathbb{P}^N$, and $X = V(F_1, \cdots, F_m)$ is not a complete intersection. Can one get a similarly simple formula for the genus of $X$, perhaps entirely in terms of the degrees of the $F_i$? Is this too much to ask? I can't even simply describe the canonical divisor.</p>