Geproci sets and the combinatorics of skew lines in $\mathbb P^3$

Geproci sets of points in $\mathbb P^3$ are sets whose general projections to $\mathbb P^2$ are complete intersections. The first nontrivial geproci sets came from representation theory, as projectivizations of the root systems $D_4$ and $F_4$. In most currently known cases geproci sets lie on very special unions of skew lines and are known as half grids. For this important class of geproci sets we establish fundamental connections with combinatorics, which we study using methods of algebraic geometry and commutative algebra. As a motivation for studying them, we first prove Theorem A: for a nondegenerate $(a,b)$-geproci set $Z$ with $d$ being the least degree of a space curve $C$ containing $Z$, that if $d\leq b$, then $C$ is a union of skew lines and $Z$ is either a grid or a half grid. We next formulate a combinatorial version of the geproci property for half grids and prove Theorem B: combinatorial half grids are geproci in the case of sets of $a$ points on each of $b$ skew lines when $a\geq b-1\geq 3$. We then introduce a notion of combinatorics for skew lines and apply it to the classification of single orbit combinatorial half grids of $m$ points on each of 4 lines. We apply these results to prove Theorem C, showing, when $n\gg m$, that half grids of $m$ points on $n$ lines with two transversals must be very special geometrically (if they even exist). Moreover, in the case of skew lines having two transversals, our results provide an algorithm for enumerating their projective equivalence classes. We conjecture there are $(m^2-1)/12$ equivalence classes of combinatorial $[m,4]$-half grids in the two transversal case when $m>2$ is prime.