# Quasi complete intersections and global Tjurina number of plane curves

## TL;DR

This paper studies quasi complete intersections in the projective plane, providing bounds on their degree based on syzygies, and recovers known results on the Tjurina number of plane curves.

## Contribution

It establishes bounds on the degree of quasi complete intersections using syzygies and generalizes results related to the Tjurina number of plane curves.

## Key findings

- Bounds on deg(T) in terms of a, b, c, and syzygy degree r
- Recovery of du Plessis-Wall theorem on Tjurina number
- Extension of results to related invariants of plane curves

## Abstract

A closed subscheme of codimension two $T \subset P^2$ is a quasi complete intersection (q.c.i.) of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O} (-a) \oplus \mathcal{O} (-b) \oplus \mathcal{O} (-c) \to \mathcal{I} _T$. We give bounds on deg$(T)$ in function of $a,b,c$ and $r$, the least degree of a syzygy between the three polynomials defining the q.c.i. As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves and some other related results.