# Geometry of quintics in $\mathbb P^3$ and the Craighero-Gattazzo surface of general type

## TL;DR

This paper investigates the properties of the tri-canonical system on the Craighero-Gattazzo surface and examines the non-rationality of certain normalized quotient curves in a complex geometric setting involving singular quintics.

## Contribution

It provides new insights into the base point freeness and tangent separation of the tri-canonical system, and establishes non-rationality results for specific quotient curves in algebraic geometry.

## Key findings

- Tri-canonical system's base point freeness analyzed
- Conditions for tangent vector separation determined
- Non-rationality of certain quotient curves proven

## Abstract

In this paper we study the question whether the tri-canonical system on the Craighero-Gattazzo surface is base point free and at which points does it separate tangent vectors. Also we study the non-rationality of the normalization of the quotient of a general curve (under a given involution) in the product linear system of cubics and quadrics on a singular quintic in $\PR^3$ with four elliptic singularities.