The non-Lefschetz locus of conics

A graded Artinian algebra $A$ has the Weak Lefschetz Property if there exists a linear form $\mathrm{\ell }$ such that the multiplication map by $\mathrm{\ell }:[A{]}_i\to [A{]}_{i+1}$ 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 $A=k[x_1,x_2,x_3]\mathrm{/}(f_1,f_2,f_3)$, with $\text{char }k=0$, has the Strong Lefschetz Property at range $2$, i.e. there exists a linear form $\mathrm{\ell }\in [R{]}_1$, such that the multiplication map $\times {\mathrm{\ell }}^2:[M{]}_i\to [M{]}_{i+2}$ has maximum rank in each degree. Then we focus on the forms of degree 2 such that $\times C:[A{]}_i\to [A{]}_{i+2}$ 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 $A=k[x_1,x_2,x_3]\mathrm{/}(f_1,f_2,f_3)$ has the expected codimension as a subscheme of ${\mathbb{P}}^5$. 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 ${\mathbb{P}}^2$, 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 $\mathcal{E}$ is semistable with first Chern class even.