Effective theta characteristics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T08:12:34Z http://mathoverflow.net/feeds/question/40473 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40473/effective-theta-characteristics Effective theta characteristics V M 2010-09-29T14:18:02Z 2010-12-15T04:47:50Z <p>Let \$C\$ be a complex smooth projective curve of genus \$g\$ and let \$N\$ be the number of effective theta-characteristics of \$C\$, or equivalently, the number of points of order two on the theta divisor of \$J(C)\$.</p> <p>It is known that, if \$C\$ is generic, \$N\$ is the number of the odd theta characteristics. Mumford proves that on a principally polarized abelian variety the theta divisor cannot contain all the points of order two. It follows that \$N&lt;2^{2g}\$.</p> <p>Given an arbitrary curve \$C\$, is it known a upper bound for \$N\$ depending on \$g\$?</p> http://mathoverflow.net/questions/40473/effective-theta-characteristics/40476#40476 Answer by stankewicz for Effective theta characteristics stankewicz 2010-09-29T14:31:11Z 2010-09-29T14:31:11Z <p>Well, there's a lower bound as odd theta characteristics on a canonical curve are effective, so there are at least \$2^{g-1}(2^g -1)\$ of them. Even thetas are trickier. </p> <p>Please consult Dolgachev's book: <a href="http://www.math.lsa.umich.edu/~idolga/topics.pdf" rel="nofollow">http://www.math.lsa.umich.edu/~idolga/topics.pdf</a></p> http://mathoverflow.net/questions/40473/effective-theta-characteristics/49481#49481 Answer by roy smith for Effective theta characteristics roy smith 2010-12-15T03:25:14Z 2010-12-15T04:47:50Z <p>In genus 4 it seems the maximum number of vanishing even theta nulls is 10, which in fact occurs on a unique 4 dimensional principally polarized abelian variety. A bound may be obtained by considering the effect on the degree of the Gauss map of the theta divisor. </p> <p>You may consult the paper of Robert Varley: <a href="http://www.jstor.org/pss/2374519" rel="nofollow">http://www.jstor.org/pss/2374519</a></p> <p>oops these are perhaps the isolated singularities on theta. I have not checked but the non isolated case of hyperelliptic jacobians may be different. Lets see, a h.e jacobian of genus 4 occurs as a double cover of P^1 branched at 10 points, so there are I guess, gosh again it seems there are 10 of them, i.e. the hyperelliptic line bundle plus one of the 10 ramification points.</p> <p>The ranks of the double points are all 3 in this case, and are all 4 in the previous isolated case.</p>