Bloch's conjecture on surfaces of general type with $p_g=q=0, K^2=3$ and with an involution

TL;DR

The paper addresses Bloch's conjecture for smooth minimal surfaces of general type with $p_g=0=q$ and $K^2=3$ that admit an involution $i$ with a bicanonical map not composed with $i$, where the quotient $S/i$ is birational to an Enriques surface or a surface of Kodaira dimension one. It adopts Voisin's Roitman-finiteness framework, studying the correspondence $\Delta_S-\mathrm{Gr}(i)$ and exploiting Prym varieties attached to ramified double covers of curves in a linear system to bound the anti-invariant part of ${\rm CH}_0(S)$. Through a case analysis of ramification loci and dimension arguments, it proves that ${\rm CH}_0(S)^-$ is finite-dimensional in the Roitman sense, which implies the involution acts trivially on ${\rm CH}_0(S)_{hom}$ in these settings. The results extend known instances of Bloch's conjecture to a new family of surfaces, and the constructions align with Ri's bi-double cover framework, broadening the scope of applicable techniques for algebraic cycles on surfaces.

Abstract

In this short note we prove that an involution on certain examples of surfaces of general type with $p_g=0=q, K^2=3$, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is bi-rational to an Enriques surface or to a surface of Kodaira dimension one and show that the Bloch conjecture holds for such surfaces.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4