Quasi-complete intersections and global Tjurina number of plane curves

A closed subscheme of codimension two T ⊂ P 2 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 ) → 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. (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.