A family of surfaces with and Albanese map of degree 3

Abstract

We study a family of surfaces of general type with pg=q=2 and K2=7, originally constructed by Cancian and Frapporti by using the Computer Algebra System MAGMA. We provide an alternative, computer‐free construction of these surfaces, that allows us to describe their Albanese map and their moduli space.

On semi‐isogenous mixed surfaces

“…The first example of a minimal surface of general type with these invariants appeared very recently: in the author constructs such surface as double cover of an abelian surface. Pignatelli and Polizzi, in the recent paper , show that our surface is different from Rito's one, proving that in our case the Albanese map is a generically finite triple cover.…”

Zero-cycles on Cancian–Frapporti surfaces

“…Theorem 2.1 (Cancian-Frapporti [7], Pignatelli-Polizzi [26]). There exist minimal surfaces S of general type with p g (S) = q(S) = 2 and K 2 S = 7, and surjective Albanese map (of degree 3).…”

“…Let S be a Cancian-Frapporti surface. Then conjecture 1.1 is true for S. This is proven by exploiting the facts that Cancian-Frapporti surfaces have (a) finite-dimensional motive (in the sense of [14]) and (b) surjective Albanese morphism [26]. A key ingredient of the argument is a strong form of the generalized Hodge conjecture for self-products of abelian surfaces [1], [32].…”

A Pair of Rigid Surfaces with pg= q = 2 and K2= 8 Whose Universal Cover is Not the Bidisk

“…Recently, R. Pignatelli and the first author studied some surfaces with p g = q = 2 and K 2 S = 7, originally constructed in [CF18] and arising as triple covers S −→ A branched over D A , where (A, D A ) is as in the previous example. We refer the reader to [PP17] for more details.…”