Geometry of dots and ropes

Abstract:

An $\alpha$-dot is the first infinitesimal neighbourhood of a point with respect to an $(\alpha - 1)$-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let $\Gamma$ be a collection of $d$ $\alpha$-dots in uniform position in ${\mathbb {P}^n},\alpha \geqslant 2$. Then the Hilbert function ${h_\Gamma }$ of $\Gamma$ satisfies \[ {h_\Gamma }(r) \geqslant \min (rn + 1,2d) + (\alpha - 2)\min ((r - 1)n - 1, d)\] for $r \geqslant 3$. Equality occurs for some $r$ with $rn + 2 \leqslant 2d$ if and only if ${\Gamma _{{\text {red}}}}$ is contained in a rational normal curve $C$, and the tangent directions to this curve at these points are all contained in $\Gamma$. Equality occurs for some $r$ with $(r - 1)n \leqslant d$ if and only if $\Gamma$ is contained in the first infinitesimal neighbourhood of $C$ with respect to a subbundle, of rank $\alpha - 1$ and of maximal degree, of the normal bundle of $C$ in ${\mathbb {P}^n}$. This implies an upper bound on the degree of a subbundle of rank $\alpha - 1$ of the normal bundle of an irreducible nondegenerate smooth curve of degree $d$ in ${\mathbb {P}^n}$, by a Castelnuovo argument.