Algebraic Geometry Seminar: The non-Lefschetz locus, jumping lines and conics, by Emanuela Marangone
September 18, 2023
3:00 PM - 3:50 PM
Emanuela Marangone (Notre Dame): The non-Lefschetz locus, jumping lines and conics
An Artinian Algebra $A$ has the Weak Lefschetz Property (WLP) if there
is a linear form, $\ell$, such that the multiplication map $\times \ell$ from $A_i$ to $A_{i+1}$ has
maximal rank for each integer $i$. We want to study the set of linear forms
for which maximal rank fails, this is called the non-Lefschetz locus and has
a natural scheme structure.
An important result by BoijMiglioreMiro-RoigNagel 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 $\times C$ from $A_i$ to $A_{i+2}$ has maximal rank
for each integer $i$. We will show that for a general complete intersection of
height 3, the non-Lefschetz locus of conics has the expected codimension as
a subscheme of $\mathbb{P}^5$, 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 $M = H^1_{?}(\mathbb{P}^2,E)$ 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.
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Date posted
Sep 21, 2023
Date updated
Sep 21, 2023
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