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Gröbner Bases for Spaces of Quadrics of Low Codimension

✍ Scribed by Aldo Conca

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
125 KB
Volume
24
Category
Article
ISSN
0196-8858

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✦ Synopsis

Let R = ⊕ i≥0 R i be a quadratic standard graded K-algebra. Backelin has shown that R is Koszul provided dim R 2 ≤ 2. One may wonder whether, under the same assumption, R is defined by a Gröbner basis of quadrics. In other words, one may ask whether an ideal I in a polynomial ring S generated by a space of quadrics of codimension ≤2 always has a Gröbner basis of quadrics. We will prove that this is indeed the case with, essentially, one exception given by the ideal I = x 2 xy y 2xz yz ⊂ K x y z . We show also that if R is a generic quadratic algebra with dim R 2 < dim R 1 then R is defined by a Gröbner basis of quadrics.

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