Quasi complete intersections and global Tjurina number 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.