Discriminants and the monoid of quadratic rings

We consider the natural monoid structure on the set of quadratic rings over an arbitrary base scheme and characterize this monoid in terms of discriminants.Comment: 23 pages; many changes made according to feedbac

Nondegenerate curves of low genus over small finite fields

In a previous paper, we proved that over a finite field $k$ of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F_2 and one over F_3. Both of these curves have remarkable extremal properties concerning the number of rational points over various extension fields.Comment: 8 pages; uses pstrick

On explicit descent of marked curves and maps

We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to D\`ebes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.Comment: 35 page