Rational divisors in rational divisor classes

We discuss the situation where a curve C, defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When C has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down to earth, approach, which indicates how to compute this divisor in certain situations. We also discuss examples where C does not have points everywhere locally, and where no such K-rational divisor is contained in the K-rational divisor class

Tau functions, Prym-Tyurin classes and loci of degenerate differentials

We study the rational Picard group of the projectivized moduli space $P\overline{{\mathfrak {M}}}_{g}^{(n)}$ of holomorphic $n$-differentials on complex genus $g$ stable curves. We define $n-1$ natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of $n$-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve

Gorenstein liaison of divisors on standard determinantal schemes and on rational normal scrolls

AbstractLet C⊂Pn be an arithmetically Cohen–Macaulay subscheme. In terms of Gorenstein liaison it is natural to ask whether C is in the Gorenstein liaison class of a complete intersection. In this paper, we study the Gorenstein liaison classes of arithmetically Cohen–Macaulay divisors on standard determinantal schemes and on rational normal scrolls. As main results, we obtain that if C is an arithmetically Cohen–Macaulay divisor on a “general” arithmetically Cohen–Macaulay surface in P4 or on a rational normal scroll surface S⊂Pn, then C is glicci (i.e. it belongs to the Gorenstein liaison class of a complete intersection)