Extending infinitely many times arithmetically Cohen-Macaulay and Gorenstein subvarieties of projective spaces

TL;DR

This paper constructs examples of infinitely extendable arithmetically Cohen-Macaulay and Gorenstein subvarieties in projective spaces that are not cones or complete intersections, using Hilbert scheme dimension computations.

Contribution

It provides new examples of infinite extensions of certain algebraic subvarieties, expanding understanding of their geometric properties beyond classical cases.

Findings

Examples of infinitely extendable ACM subvarieties

Examples of infinitely extendable Gorenstein subvarieties

These examples are not cones or complete intersections

Abstract

We give examples of infinitely extendable (not as cones) arithmetically Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension $2$ subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$ subschemes due to J. O. Kleppe and R.-M. Mir\'{o}-Roig.