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 :[A]ᵢ→ [A]ᵢ₊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=k[x₁,x₂,x₃]/(f₁,f₂,f₃), with char k=0, has the Strong Lefschetz Property at range $2$, i.e. there exists a linear form ∈ [R]₁, such that the multiplication map × ²: [M]ᵢ→ [M]ᵢ₊₂ has maximum rank in each degree. Then we focus on the forms of degree 2 such that × C: [A]ᵢ→ [A]ᵢ₊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=k[x₁,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.