Normal cones of monomial primes

We explicitly calculate the normal cones of all monomial primes which define the curves of the form (tL, tL+1, . . . , tL+n), where n ≤ 4. All of these normal cones are reduced and Cohen-Macaulay, and their reduction numbers are independent of the reduction. These monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed. Substantial use was made of the computer algebra packages Maple and Macaulay2. Let (R,m) be a regular local or graded local ring and let I ⊆ R be an ideal. In the case of a graded ring, I is assumed to be homogeneous. By NI = ⊕ t∈N I /m ·I we denote the special fibre of the blow-up of I, i.e., the normal cone of I. When I ⊆ R is an m–primary ideal (in which case Spec(NI) is homeomorphic to the exceptional fibre of the blow–up of I), normal cones have been studied quite intensely (see for example the comprehensive reference [HIO]), and some of the results for m-primary ideals have been extended to equimultiple ideals (cf. [Sh1], [Sh2], [HSa], [CZ]). For more general ideals very little is known about the structure of their normal cones. If I is generated by a d-sequence, then NI is a polynomial ring, cf. [Hu]. In particular this is the case if I is generated by a regular sequence. Conversely, a celebrated result of Cowsik and Nori [CN] asserts that an equidimensional radical ideal I with dim(NI) = ht(I) = dim(R) − 1 is a complete intersection ideal. Other than that, the structure of NI has been determined only in some special cases ([MS], [G], [CZ]). Our interest in the normal cones got sparked by their relations to evolutions and evolutionary stability of algebras. In [H] it was shown that whenever I is a radical ideal and NI is reduced (or, more generally, NI does not contain any nilpotent elements of degree 1), then the ideal m · I is integrally closed. If in addition R is essentially of finite type over a field k of characteristic zero, this in turn implies that R/I is evolutionarily stable as a k-algebra, thus answering a question of Mazur from [EM] in this case. In [HH] further connections between the normal cone of an ideal and the integral closedness of m · I have been established. For two-dimensional regular rings the product of any two integrally closed ideals is integrally closed; however, in general this is not the case. With the exception of radical ideals generated by d-sequences, few examples of classes of ideals I for Received by the editor February 22, 2000 and, in rvised form, February 28, 2001. 2000 Mathematics Subject Classification. Primary 13-04, 13C14.