Varieties of minimal degree in weighted projective space

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.