IJPAM: Volume 40, No. 3 (2007)

ON THE HILBERT FUNCTIONS OF DISJOINT LINES
IN ${\bf {P}}^n$ AND OF THEIR SPANNING SUBSETS

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: [email protected]


Abstract.Here we discuss the following questions. Fix integers $n \ge 3$ and $x >0$. Let $\{P_i,Q_i\}$, $1 \le i \le x$, $x$ non-ordered set of unordered pairs of distinct points of ${\bf {P}}^n$. Set $S:= \cup _{i=1}^{x} \{P_i,Q_i\}$, $D_i:= \langle \{P_i,Q_i\}\rangle$ and $X:= \cup _{i=1}^{x} D_i \subset {\bf {P}}^n$. Assume that $X$ is the union of $x$ distinct lines, i.e. assume $\sharp (S) = 2x$, $\{P_j,Q_j\}\cap D_i = \emptyset$ for all $i \ne j$ and that no plane contains $\{P_i,Q_i,P_j,Q_j\}$ for some $i \ne j$. What can be said about the Hilbert function $h_X$ of $X$ in terms of the Hilbert function $H_S$ of $X$ (and viceversa)? Can the base locus of $\vert \mathcal {I}_S(t)\vert$ contain a line?

Received: June 10, 2007

AMS Subject Classification: 14N05

Key Words and Phrases: disjoint lines, lines in projective spaces, Hilbert function, postulation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 40
Issue: 3