# Special components of Noether-Lefschetz loci

## TL;DR

This paper investigates special components of Noether-Lefschetz loci related to certain algebraic curves on degree 8 surfaces, providing evidence for their structure, maximum codimension, and potential counterexamples to Harris's conjecture.

## Contribution

It offers a detailed analysis of the structure and intersections of Noether-Lefschetz loci for specific cohomology classes, and proposes a conjectural counterexample to Harris's conjecture.

## Key findings

- Noether-Lefschetz loci are distinct for most parameters
- Maximum codimension of these loci is 35
- Existence of infinite components passing through Fermat surfaces

## Abstract

We take a sum $C_1+r C_2,\ r\in\mathbb Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside a smooth surface of degree $8$ and with $C_1\cap C_2=\emptyset$. We gather evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+ r C_2$ are distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a conjectural description of a counterexample to a conjecture of J. Harris. The methods used in this paper also produce in a rigorous way an infinite number of general components passing through the point representing the Fermat surface of degree $\leq 9$, and many non-reduced components for such degrees.