On the type and generators of monomial curves

Nguyen Thi DUNG

doi:10.3906/mat-1708-16

Abstract

Let $n_1, n_2,\ldots, n_d$ be positive integers and $H $ be the numerical semigroup generated by $n_1,n_2, \ldots, n_d$. Let $A:=k[H]:=k[t^{n_1}, t^{n_2},\ldots, t^{n_d}]\cong k[x_1,x_2,\ldots,x_d]/I$ be the numerical semigroup ring of $H $ over $k.$ In this paper we give a condition $(*)$ that implies that the minimal number of generators of the defining ideal $I$ is bounded explicitly by its type. As a consequence for semigroups with $d=4$ satisfying the condition $(*)$ we have $\mu ({\rm in}(I))\leq 2(t(H))+1$.

Highlights

We say that the semigroup H (or the ring k[H]) satisfies the condition (*) if for every binomial Mi − Ni ∈ G(H) , with Ni ≺degrevlex Mi , the variable x1 divides the monomial Ni for all i
Let n1 < n2 < . . . < nd be positive integers such that gcd(n1, n2, . . . , nd) = 1 and H = ⟨n1, n2, . . . , nd⟩ = {∑di=1 cini | ci ∈ N for all 1 ⩽ i ⩽ n} be the numerical semigroup minimally generated by n1, n2, . . . , nd, where N stands for the set of nonnegative integers
If we set I := I(H) the kernel of the graded ring homomorphism Φ : R → A defined by Φ(xi) = tni for each 1 ⩽ i ⩽ d, the ring A ⊂ k[t] has a presentation as a quotient R/I and I is called the defining ideal of H

Summary

Introduction

We say that the semigroup H (or the ring k[H]) satisfies the condition (*) if for every binomial Mi − Ni ∈ G(H) , with Ni ≺degrevlex Mi , the variable x1 divides the monomial Ni for all i . Example 2.4 (i) Let R = k[x, y] be a polynomial ring of 2 variables over k and monomial ideal J = (x6, x5y2, x2y4, y6)R .

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Conclusion