Thomae-Weber Formula: Algebraic Computations of Theta Constants

We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete level 2 structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.

THE CURVE OF LINES ON A PRIME FANO THREE-FOLD OF GENUS 8

We show that a general prime Fano three-fold X of genus 8 can be reconstructed from the pair (Γ, L), where Γ is its Fano curve of lines and L = OΓ(1) is the theta-characteristic which gives a natural embedding Γ ⊂ ℙ5.