The Rees Algebra for Certain Monomial Curves

TL;DR

This paper explicitly determines the equations of the Rees algebra for specific monomial curves and proves that their blowup schemes are not smooth, confirming Francia's conjecture for certain prime ideals.

Contribution

It provides explicit equations for the Rees algebra of certain monomial curves and verifies Francia's conjecture regarding smooth blowups of prime ideals.

Findings

Rees algebra equations for specific monomial curves are explicitly derived.

The blowup scheme of these curves is shown to be non-smooth.

Confirms Francia's conjecture for certain prime ideals in regular local rings.

Abstract

In this article, we find the equations defining the Rees algebra for certain Monomial Curves explicitly and use them to prove that the blowup scheme is not smooth. This proves a conjecture of Francia in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.