2024/04/24 by Emanuela Marangone, Marangone, Emanuela
#math.AC #math.AG
paper · pdf · doi:10.48550/arxiv.2404.16238
A graded Artinian algebra A has the Weak Lefschetz Property if there exists a linear form ℓ such that the multiplication map by ℓ:[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 ℓ∈ [R]1, such that the multiplication map × ℓ2: [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.