# Algebraic cycles and Verra fourfolds

## TL;DR

This paper investigates the Chow ring of Verra fourfolds, demonstrating that homologically trivial 1-cycles are generated by conics and establishing a multiplicative Chow-K"unneth decomposition with implications for intersection products.

## Contribution

It proves that the Chow group of homologically trivial 1-cycles on general Verra fourfolds is generated by conics and introduces a multiplicative Chow-K"unneth decomposition for these fourfolds.

## Key findings

- Homologically trivial 1-cycles are generated by conics.
- Verra fourfolds admit a multiplicative Chow-K"unneth decomposition.
- Implications for intersection product structure in the Chow ring.

## Abstract

This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-K\"unneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.