Intersection of curves in projective 4 space
Luca Chiantini, Łucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg, Justyna Szpond
TL;DR
This work extends intersection bounds for two reduced irreducible curves from $\mathbb{P}^3$ to $\mathbb{P}^4$ by introducing the bounds $$B(d_1,d_2)$ and $$B_g(d_1,d_2)$$ and by leveraging genus calculations tied to the $h$-vector of a general hyperplane section. It proves sharp bounds for curves on a cubic surface (Theorem A) and constructs explicit examples achieving these bounds, including on the del Pezzo quartic surface, while also deriving low-degree results (Proposition C) showing quintic curves lie on cubics. The paper then develops a systematic framework—via the Hartshorne–Rao module, lifting lemmas, and meticulous $h$-vector analysis—to compare $B_g$ with $B$, obtaining partial confirmations of the conjectured global bound in many cases (notably when at least one curve is ACM). Collectively, these results sharpen the Diaz–Giuffrida type bounds in $\mathbb{P}^4$, reveal when equality can occur, and lay groundwork for further exploration in higher dimensions and broader curve classes.
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 Miró-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 initial results in dimensions greater than 3. Here we continue this study for dimension 4. We introduce a number B defined in terms of the degrees of the curves and prove that when both curves lie on a surface of minimal degree (thus a cubic surface) then the number of points of intersection is at most B. Moreover, we conjecture that B is always an upper bound and we prove this conjecture in many cases, including when at least one of the curves is ACM. Our approach focuses on the genera of the curves and their union. In addition we define a second number B' in terms of the degrees and the genus of the union which we can show bounds the number of points of intersection above, and we use a variety of methods to study how B and B' compare.