Varieties of minimal degree in weighted projective space

TL;DR

This paper explores varieties of minimal degree within weighted projective spaces, establishing bounds, defining weighted determinantal scrolls, and analyzing their properties and differences from classical cases.

Contribution

It introduces a framework for studying minimal degree varieties in weighted projective spaces, including bounds, weighted determinantal scrolls, and regularity properties.

Findings

Sharp bounds for minimal degree subvarieties in divisible weighted projective spaces.

Definition and analysis of weighted determinantal scrolls and their properties.

Characterization of minimal degree conditions and weighted $N_p$ properties.

Abstract

We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate subvariety of a divisible weighted projective space has minimal degree. We define a weighted notion of $1$-generic matrices and, in analogy with the classical theory, show that there is a theory of weighted determinantal scrolls. Moreover, we characterize precisely when these have minimal degree and determine their weighted $N_p$ properties, and tie this to two weighted notions of regularity. Finally, we propose conjectural bounds for more general weighted threefolds and pose several natural questions. Throughout, we highlight the differences between this theory and the classical case.