Intersection of curves in projective 4 space
Luca Chiantini, {\L}ucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg, Justyna Szpond

TL;DR
This paper investigates the maximum intersection points of two algebraic curves in projective 4-space, extending known results from lower dimensions and introducing new bounds and conjectures based on the curves' properties.
Contribution
It introduces a new bound B for intersection points when curves lie on minimal degree surfaces in 4-space and proves the conjecture in cases including when one curve is ACM.
Findings
Bound B is established for curves on cubic surfaces in 4-space.
Conjecture that B always bounds the intersection points is proven in many cases.
A second bound B' based on genus and degree is also introduced and compared.
Abstract
Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space is the plane, this is trivial. For projective 3 space this problem was solved independently by Diaz and by Giuffrida in 1986. They showed that two curves achieving the maximum number of intersection points have to be rational curves on a smooth surface of minimal degree, i.e., a quadric surface. Note that these curves are far from being arithmetically Cohen-Macaulay. In contrast, Hartshorne and Mir\'o-Roig addressed this problem in 2015 for space curves under the assumption that the curves are arithmetically Cohen-Macaulay (ACM), introducing very deep techniques and obtaining very different results from Diaz and Giuffrida. Diaz and Giuffrida also gave…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Tensor decomposition and applications
