The non-Lefschetz locus of conics

A graded Artinian algebra A has the Weak Lefschetz Property if there exists a linear form such that the multiplication map by: Aᵢ A₈+₁ has maximum rank in every degree. The linear forms satisfying this property form a Zariski-open set; its complement is called the non-Lefschetz locus of A. In this paper, we investigate analogous questions for degree-two forms rather than lines. We prove that any complete intersection A=kx₁, x₂, x₃/ (f₁, f₂, f₃), with char k=0, has the Strong Lefschetz Property at range 2, i. e. there exists a linear form R₁, such that the multiplication map ²: Mᵢ M₈+₂ has maximum rank in each degree. Then we focus on the forms of degree 2 such that C: Aᵢ A₈+₂ fails to have maximum rank in some degree i. The main result shows that the non-Lefschetz locus of conics for a general complete intersection A=kx₁, x₂, x₃/ (f₁, f₂, f₃) has the expected codimension as a subscheme of P⁵. The hypothesis of generality is necessary. We include examples of monomial complete intersections in which the non-Lefschetz locus of conics has different codimension. To extend a similar result to the first cohomology modules of rank 2 vector bundles over P², we explore the connection between non-Lefschetz conics and jumping conics. The non-Lefschetz locus of conics is a subset of the jumping conics. Unlike the case of the lines, this can be proper when E is semistable with first Chern class even.