Putting points on curves
Algebraic Geometry Seminar
Putting points on curves
Series: Algebraic Geometry Seminar
Location: MATH 501
Presenter: Arvind Suresh, Department of Mathematics, University of Arizona
Fix a finite extension of fields L/K of degree n. We present a Galois-cohomological approach to finding genus g curves X/K having a closed point with residue field L. There is a coarse moduli space M_{g,L/K} for this problem, which is a twisted form of the classical moduli space M_{g,n} which parametrizes n-pointed genus g curves. Our strategy basically consists in twisting a given G-equivariant morphism V --> M_{g,n} (with G a subgroup of S_n and V/K a variety having many rational points) to get a similar morphism of twisted forms V' --> M_{g,L/K}. We discuss some limitations with this idea, and also survey some of the known results about the spaces M_{g,L/K}.