Affine curves in characteristicp are set theoretic complete intersections

Abstract

We prove here that any curve in the affine n-space over a field k of positive characteristic p is a set theoretic complete intersection. Szpiro proved that a curve which is a local complete intersection in affine 3-space is a set theoretic complete intersection. See, Ferrand [1], Szpiro [-2]. A consequence of Mohan Kumar's paper [3] is that any local complete intersection curve in any affine nspace is a set theoretic complete intersection. We here first prove the result in the complete local case and use that to prove the general result. [ (b) If S is the multiplicative set of nonzero divisors in A, then S-1A = S-1 B, i.e. SpecB~SpecA is birational isomorphism.Let C be the conductor of the inclusion of A in B. Then C is primary with respect to the unique maximal ideal of B, and therefore there is a positive integer N such that X~'" ~ C for 3-<j < n, where )(j denotes the image of Xj in B; therefore, there are hsek [[X1, Xz]]~uch that X~"-hj~l for 3<j<n. Our first claim is:(A) The ideal J generated by g and X~"-hj, 3 <j<n, has its radical equal to 1. Now, because S- us, vjek[[X1,Xz]], uj relatively prime to g, such that X.u.-v.~I, for 3<j<n. We shall show that X,u,-v,~radical J. Put a= [XJjusZvs) ,~, b 88 N,we have a=b +c. Now a and b belong to I, and therefore ceI. But csk [[Xi,X2]], so that g divides c, and consequently c e J. Now b, c e J, proving that a s J and that Xjuj -vjeradJ.

Almost complete intersection monomial curves inA4

“…Since F+(u2 + v3) = X~"~-~~(~~) X , " " -"~' "~) , we may assume that it is divided by F-(w4). Hence ui ( v 4 ) 2, ai ( v 4 ) < 0 for i = 3,4. Lemma 2.8 Let d = g.c.d.…”

“…It is natural to ask whether all curves are set-theoretic complete intersections, that is, there is a complete intersection ideal which is equal up to radical. Cowsik and Nori [3] showed that every affine curve in characteristic p is set-theoretic complete intersection. Related to monomial curves, it is known that it is true in the following cases;…”

Affine monomial curves

“…The conjecture is true if charactertistic of k is a prime, by the famous result proved by Cowsik and Nori [12]. However, very little is known in characteric zero, except for some special cases.…”

Radically perfect prime ideals in polynomial rings

“…This question, apparently, was first considered by Kronecker in late 19th century. Since then an enormous amount of research has evolved around these types of questions (see, e.g., [1,2], [5, Chapter V], [6] and the references there in. Note that in these references the phrase "set theoretic complete intersection" is used for radically perfectness).…”

“…Among them still remaining unsolved is the question whether height two ideals in K[X, Y, Z] are radically perfect, where K is a field of characteristic zero. When K is of positive characteristic, it is known that height n − 1 ideals in the polynomial ring in n variables over K are radically perfect [1]. In search of an answer to the characteristic zero case, we consider the following question.…”