Liftings of a monomial curve

TL;DR

This paper introduces a lifting operation on monomial curves, showing that most liftings have Cohen-Macaulay tangent cones and that Betti sequences stabilize under this process, revealing structural invariants.

Contribution

It demonstrates that all but finitely many liftings of a monomial curve have Cohen-Macaulay tangent cones, and Betti sequences become eventually constant under lifting.

Findings

Most liftings have Cohen-Macaulay tangent cones.

Betti sequences stabilize under repeated lifting.

Liftings preserve Cohen-Macaulay property when original curve has it.

Abstract

We study an operation, that we call lifting, creating non-isomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen-Macaulay tangent cones even if the tangent cone of the original curve is not Cohen-Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen-Macaulay tangent cones when the original monomial curve has a Cohen-Macaulay tangent cone. In this case, all the Betti sequences are nothing but the Betti sequence of the original curve.