On the Multigraded Hilbert Function of Lines and Rational Curves in Multiprojective Spaces

Abstract

We study the multigraded Hilbert function of general configurations of lines in multiprojective spaces. In several cases we prove that this multigraded Hilbert function is the expected one. We make conjectures about other configurations and for small genus curves with a prescribed multidegree.

Interpolation for Brill–Noether curves in $${\mathbb {P}}^4$$

“…Since it is 2 conditions for a surface in P 4 to pass through a point and 1 condition to meet a line, S cannot pass through 9 general points points while meeting a general line. This contradicts (3), which predicts that C should be able to pass through 9 general points points while meeting a general line. If ( d, g , r ) = (7 , 2 , 5) : Then r 2 + 2 r − 6 = 29 .…”

“…As in previous eras, this attention has been motivated by both intrinsic interest, as well as by striking applications to a wide range of other interesting geometric problems. Recent examples of such problems include smoothing curve singularities [27], constructing moving curves in M g [2,6], the first author's resolution of Severi's 1915 Maximal Rank Conjecture [20], as well as various generalizations thereof [3].…”