User jonah - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:40:26Z http://mathoverflow.net/feeds/user/10054 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42127/generalization-of-bezouts-theorem generalization of Bezout's Theorem? Jonah 2010-10-14T07:46:38Z 2010-11-01T04:12:45Z <p>I learned Bezout's Theorem in class, stated for plane curves (if irreducible, sum of intersection multiplicities equals product of degrees). What is the proper general statement, for projective varieties of degree n?</p> <p>I think it is something like: If finite, the sum of multiplicities equals the product of degrees.. else the (dimension? degree? sums over irreducible components?) of the intersection is less than or equal to the difference in degrees.</p> <p>Help is appreciated!</p> http://mathoverflow.net/questions/55014/some-questions-on-riemann-surface/55022#55022 Comment by Jonah Jonah 2011-02-12T09:05:33Z 2011-02-12T09:05:33Z Sure. Say the automorphism of the surface you start with is called f, and the lifted automorphism is called F, and pick x in H. Pick a small enough neighborhood of F(x) so that the restriction of the (holomorphic) covering map is an analytic isomorphism. Now it is clear, since f compose pi is the same as pi compose F, that F can be written, in a suitably small neighborhood of x, as the composition of holomorphic maps. http://mathoverflow.net/questions/55014/some-questions-on-riemann-surface/55022#55022 Comment by Jonah Jonah 2011-02-11T16:48:17Z 2011-02-11T16:48:17Z For Q3: If H is the upper half-plane, S the Riemann surface, and pi:H-&gt;S, then composing pi with any automorphism gives a holomorphic map from H to S. Since this map takes a simply-connected surface to S (in particular the image of pi_1 of H is trivial), it must lift to a holomorphic map to H.