Speaker: Juan Migliore
University of Notre Dame
Will give an Algebraic Geometry and Commutative Algebra Seminar entitled:
Intersections of curves in P^4
Abstract: I'll speak on joint work (almost finished) with Luca Chiantini (Siena), \L ucja Farnik (Krakow), Giuseppe Favacchio (Palermo), Brain Harbourne (Nebraska), Tomasz Szemberg (Krakow) and Justyna Szpond (Krakow). In how many points can two irreducible, non-degenerate curves, $C_1$ and $C_2$, of degrees $d_1$ and $d_2$ respectively, meet in projective space? For $\mathbb P^3$ we have a pretty good picture, thanks to work of Diaz (1986) and of Giuffrida (1986). Refinements were made by Hartshorne--Mir\'o-Roig (2015) requiring the curves to be arithmetically Cohen-Macaulay, and by Chiantini-Migliore (2021) allowing the curves to be reducible. According to Hartshorne and Mir\'o-Roig, ``There seems to be scant attention to these questions in the literature.'' The Diaz-Giuffrida bound is $(d_1-1)(d_2-1) +1$ and occurs exactly when one curve is of type $(d_1-1,1)$ on a smooth quadric surface $Q$ (a surface of minimal degree), and $C_2$ is of type $(1,d_2-1)$ on the same $Q$. If the curves do not both lie on a smooth quadric, the curves cannot achieve the bound (there are open questions about this). In the current project we ask the analogous question for curves in $\mathbb P^4$, and we get some surprising similarities and differences from the situation in $\mathbb P^3$, which will be described in this talk. After a computation for intersections of curves on a cubic surface (a surface of minimal degree), our approach turns to a study of the arithmetic genus of $C_1 \cup C_2$ and a very careful study of the Hilbert function of the general hyperplane section, $\Gamma$, of $C_1 \cup C_2$. Along the way we need to ``lift" properties of $\Gamma$ to $C_1 \cup C_2$, which we do using a theorem of Huneke-Ulrich and Strano.
Date: 11-08-2024
Time: 3:00 pm
Location: 258 Hurley Bldg
Originally published at math.nd.edu.