On generically polarized Gorenstein surfaces of sectional genus two.
It is known that generators of ideals defining projective toric varieties of dimension embedded by global sections of normally generated line bundles have degree at most . We characterize projective toric varieties of dimension whose defining ideals must have elements of degree as generators.
The moduli space M of surfaces of general type with pg = q = 1, K2 = g = 3 (where g is the genus of the Albanese fibration) was constructed by Catanese and Ciliberto in [14]. In this paper we characterize the subvariety M2 ⊂ M corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset M0 ⊂ M which parametrizes isomorphism classes of surfaces with birational bicanonical map.