PhD Defense: Emanuela Marangone - University of Notre Dame
PhD Defense: Emanuela Marangone - University of Notre Dame
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Location:125 Hayes-Healy Bldg
Speaker: Emanuela Marangone University of Notre Dame
Will give a PhD Defense entitled: The Non-Lefschetz Locus of Lines and Conics
Abstract: An Artinian Algebra A has the Weak Lefschetz Property if there is a linear form, $\ell$, such that the multiplication map $\times \ell$ has maximal rank for each degree. The non-Lefschetz locus is the set of linear forms for which maximal rank fails, and it has a natural scheme structure. Boij--Migliore--Miró-Roig--Nagel show that for a general Artinian complete intersection of height 3, the non-Lefschetz locus has the expected codimension. We want to address the same type of question for forms of degree 2 instead of lines. We define the non-Lefschetz locus for conics and show that for a general complete intersection of height 3, it has the expected codimension as a subscheme of $\mathbb P^5$. The same does not hold for certain monomial complete intersections. Later, we will generalize the study of the non-Lefschetz locus to modules $M = H^1_*(\mathbb{P}^2, \mathcal{E})$ where $\mathcal{E}$ is a vector bundle of rank 2. In this case, we show that the non-Lefschetz locus is exactly the set of jumping lines of $\mathcal E$, and the expected codimension is achieved when $\mathcal E$ 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 can be proper.
