A remark on the Chow ring of Sicilian surfaces

TL;DR

This paper introduces a conjecture relating the vanishing of a cup product map to zero intersections in the Chow group, and confirms it for Sicilian surfaces.

Contribution

It proposes a new Bloch-type conjecture for surfaces and proves it specifically for Sicilian surfaces.

Findings

Conjecture holds for Sicilian surfaces

Cup product map vanishes implies zero intersection in Chow group for these surfaces

Provides evidence for a broader conjecture in algebraic geometry

Abstract

We propose a "Bloch type" conjecture for surfaces: if the cup product map in coherent cohomology is zero, then all intersections of homologically trivial divisors should be zero in the Chow group of zero-cycles. We prove this conjecture for Sicilian surfaces.