Quasi-complete intersections and global Tjurina number of plane curves

Abstract

A closed subscheme of codimension two T⊂P2 is a quasi complete intersection (q.c.i.) of type (a,b,c) if there exists a surjective morphism O(−a)⊕O(−b)⊕O(−c)→IT. 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. (see Theorem 6). As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves (see Theorem 20) and some other related results.