In questo lavoro vengono costruite famiglie di 3-folds algebriche e non singolari di tipo generale tali che l'invariante sia il minimo possibile rispetto al genere geometrico , quando si suppone che il morfismo canonico sia birazionale. Per tali 3-folds vale la relazione lineare inoltre l'immagine del morfismo canonico é una varietà di Castelnuovo di .
Studying the connection between the title configuration and Kummer surfaces we write explicit quadratic equations for the latter. The main results are presented in Theorems 8 and 16.
Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
In this note, we consider a one-parameter family of Abelian varieties , and find an upper bound for the average rank in terms of the generic rank. This bound is based on Michel's estimates for the average rank in a one-parameter family of Abelian varieties, and extends previous work of Silverman for elliptic surfaces.
Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].