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Checking if the teoric ideal of an affine monomial curve is a complete intersection
| Content Provider | Semantic Scholar |
|---|---|
| Author | Díaz, María Isabel Bermejo Marco, Ignacio García |
| Copyright Year | 2006 |
| Abstract | Let d = {d1, . . . , dn} be a set of all-different positive integers and consider the monomial curve Γ = {(td1 , . . . , tn) ∈ Ak | t ∈ k} . The image of the homomorphism of k-algebras φ : R → k[t]; xi 7−→ tdi will be denoted by k[Γ] and its kernel will be denoted by I(d1, . . . , dn). The ideal I(d1, . . . , dn) is called the toric ideal of Γ. It is a quasi-homogeneous ideal, i.e., a homogeneous ideal when one gives degree di to variable xi for all i ∈ {1, . . . , n}. Since k[t] is integral over k[Γ] , the height of I(d1, . . . , dn) is equal to n − 1. By [10, Proposition 7.1.2], the toric ideal I(d1, . . . , dn) is generated by quasi-homogeneous binomials. According to [5, Lemma 3.4 and Remark 3.5], if either gcd(d) = 1 or k is algebraically closed, we get Γ = V (I(d1, . . . , dn)), i.e., Γ is a toric variety. If k is an infinite field, by Corollary 7.1.12 in Villarreal [10] , the ideal I(Γ) of polynomials vanishing on Γ is equal to I(d1, . . . , dn) . The prime ideal I(d1, . . . , dn) is called a complete intersection if there exists a system of quasi-homogeneous binomials g1, . . . , gn−1 such that I(d1, . . . , dn) = (g1, . . . , gn−1). |
| Starting Page | 45 |
| Ending Page | 48 |
| Page Count | 4 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ricam.oeaw.ac.at/mega2007/electronic/D.pdf |
| Alternate Webpage(s) | http://www.ricam.oeaw.ac.at/mega2007/openconf/electronic/D.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
