September 8th, 2023 – Emanuela Marangone: The non-Lefschetz locus, jumping lines and conics
An Artinian Algebra A has the Weak Lefschetz Property (WLP) if there is a linear form, ℓ, such that the multiplication map ×ℓ from Ai to Ai+1 has maximal rank for each integer i. In such case, ℓ is called Lefschetz element. When the WLP holds, we want to study the set of linear forms that are not Lefschetz elements, this is called the non-Lefschetz locus and has a natural scheme structure. An important result by Boij–Migliore–Miró-Roig–Nagel states that for a general Artinian complete intersection of height 3, the non-Lefschetz locus has the expected codimension and the expected degree.
In this talk, we will define in a similar way the non-Lefschetz locus for conics. We say that C, a homogeneous polynomial of degree 2, is a Lefschetz conic for A if the multiplication map ×C from Ai to Ai+2 has maximal rank for each integer i. We will show that for a general complete intersection of height 3, the non-Lefschetz locus has the expected codimension as a subscheme of P5, and that the same does not hold for certain monomial complete intersections.
The study of the non-Lefschetz locus for Artinian complete intersections can be generalized to modules M = H1*(P2, E) where E is a vector bundle of rank 2. The non-Lefschetz locus, in this case, is exactly the set of jumping lines of E, and the expected codimension is achieved under the assumption that E is general. In the case of conics, the same is not true. The non-Lefschetz locus of conics is a subset of the jumping conics, but it is a proper subset when E is semistable with first Chern class even.
October 6th, 2023 – Juan Ramirez: Some words on Rees algebra of edge ideals
Let R be a Noetherian ring and I an ideal. The Rees algebra of I, is defined to be the R-subalgebra, R(I) = R[It] = ⊕i≥0 Iiti, of the polynomial ring R[t]. Consider a simple graph G and its edge ideal I = I(G). In 1995, Villareal gave a description of the defining equations of the Rees algebra of edge ideals R(I(G)). In this talk, I will present this result and some conditions for when the ideal I is of linear type.
November 3rd, 2023 – Sandra Sandoval: Symbolic powers of Derksen Ideals
Given that the symbolic and ordinary powers of an ideal do not always coincide, we look for conditions on the ideal such that the equality holds for every natural number. In this talk we will define a type of ideals defined by finite groups acting linearly on a polynomial ring called Derksen ideals. We will show that for certain groups, the symbolic and ordinary powers of the Derksen ideal coincide for every natural number.
December 1st, 2023 – Lizda Moncada: The containment problem of symbolic powers vs regular powers of squarefree monomial ideals
In this talk, we explore the containment problem of symbolic and regular powers in squarefree monomial ideals, with a focus on the correlation between these powers and the properties of the associated hypergraphs. Additionally, we will give a criterion for determining when the second symbolic and ordinary powers of the edge ideal of a 3-partite hypergraph are equal.