# The double point formula with isolated singularities and canonical embeddings

## TL;DR

This paper extends Severi's double point formula to surfaces with isolated singularities, providing new criteria for canonical embeddings, especially for surfaces with geometric genus five.

## Contribution

It introduces generalized double point formulas for varieties with isolated singularities and characterizes canonical embeddings for certain surfaces with geometric genus five.

## Key findings

- Extended double point formula for surfaces with rational double points
- Canonical model embedding criteria for surfaces with p_g=5
- Characterization of embeddings as complete intersections of specific types

## Abstract

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus $p_g=5$: the canonical model is embedded in $\mathbb{P}^4$ if and only if we have a complete intersection of type $(2,4)$ or $(3,3)$.