A bound for the arithmetic genus of curves in Grassmannians

Abstract

From the introduction: Let $X\subset\bbfP^n$ be a non-degenerate degree $d$ variety over the field of complex numbers, which is ruled by $k$-planes over a curve. Let us also suppose that there is no point of $X$ such that all the rules pass through it, i.e. that $X$ is not a cone.\par We can associate to $X$ a curve $C_X$ lying in the Grassmann variety of $k$-planes in $\bbfP^n$. The goal of this note is to show that the arithmetic genus of such a curve is bounded by $\pi(d,n)$, where $\pi(d,n)$ is Castelnuovo's bound for the genus of degree $d$ curves in $\bbfP^n$ .The proof of the result relies on a previous one that establishes that the bound holds and is sharp for ruled surfaces in $\bbfP^n$. The key idea of the proof is to show that the curve in $G(k,\bbfP^n)$ associated to $X$ spans at least a $\bbfP^n$.