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2010, Bulletin of the Polish Academy of Sciences Mathematics
We give a simple geometric proof of Mohan Kumar's result about complete intersections.
2007
This paper gives a new proof of this theorem, one which mostly parallels Roberts' original proof, but avoids the use of local Chern characters. Instead, the proof here uses Adams operations on K-theory with supports as developed by Gillet-Soule.
2021
We prove the Hasse principle for a smooth projective variety X ⊂ Pn−1 Q defined by a system of two cubic forms F,G as long as n ≥ 39. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over Q.
Mathematische Annalen, 1992
Combinatorica, 1992
Let J(,d) be the set of submatrices of an m x n matrix A. Then J(M) is a ranked poset with respect to the inclusion , and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at least t? We have a solution for t = 1,2 using the followoing theorem of independent interest. Let m(n,i,j, k)= max(I,~ I + I$1), where maximum is taken for all possible pairs of families of subsets of an n-element set such that ~ is/-intersecting, ~ is j-intersecting and :~ ansd ~ are cross-k-intersecting. Then for i <j < k, m(n,i,j,k) is attained if ~ is a maximal/-intersecting family containing subsets of size at least ,~, and ~ is a maximal 2k-/-intersecting family.
Journal of Algebra, 2001
The starting point for this work is Auslander's seminal paper [A]. The main focus of his paper was to understand when the tensor product M ⊗ R N of finitely generated modules M and N over a regular local ring R is torsion-free. This condition forces the vanishing of a certain Tor module associated to M and N , which in turn, by Auslander's famous rigidity theorem, implies that Tor R i (M, N ) = 0 for all i ≥ 1. The vanishing of Tor carries a great deal of information; for example it implies the "depth formula" depth(M ) + depth(N ) = dim(R) + depth(M ⊗ R N ). From this formula one can deduce, for example, the highly non-trivial fact that if M ⊗ R N is torsion-free and non-zero, then M and N must both be torsion-free.
Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, 2016
Given a 0-dimensional subscheme X of a projective space P n K over a field K , we characterize in different ways whether X is the complete intersection of n hypersurfaces. Besides a generalization of the notion of a Cayley-Bacharach scheme, these characterizations involve the Kähler and the Dedekind different of the homogeneous coordinate ring of X or its Artinian reduction. We also characterize arithmetically Gorenstein schemes in novel ways and bring in further tools such as the module of regular differential forms, the fundamental class, and the Jacobian module of X. Throughout we strive to work over an arbitrary base field K and keep the scheme X as general as possible, thereby improving several known characterizations.
2010
This is an expository account based mainly on an article by Jack Ohm titled “Space curves as ideal-theoretic intersections”. It also gives a proof of the fact that smooth space curves can be realized as set-theoretic complete intersections. The penultimate section proves the theorem of Cowsik and Nori : Curves in affine n-space of characteristic p are set-theoretic complete intersection.
Commentationes Mathematicae Universitatis Carolinae, 2001
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Proceedings of the American Mathematical Society, 2016
Let A A be a graded complete intersection over a field and B B the monomial complete intersection with the generators of the same degrees as A A . The EGH conjecture says that if I I is a graded ideal in A A , then there should be an ideal J J in B B such that B / J B/J and A / I A/I have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.
European Journal of Combinatorics, 1988
This paper deals with intersection properties for structures extending earlier results by M. Simonovits and V. T. S6s. One of the results asserts that for a given graph G = (V, E) with girth r (satisfying some additional technical assumption), if G 1 , .
Journal of the London Mathematical Society, 1985
In this paper we discuss two results in commutative algebra that are used in A. Wiles's proof that all semi-stable elliptic curves over Q are modular . We first fix some notation that is used throughout this paper. Let O be a complete Noetherian local ring with maximal ideal m O and residue field k = O/m O . Suppose that we have a commutative triangle of surjective homomorphisms of complete Noetherian local O-algebras:
In this paper we obtain a very general intersection theorem for the values of a map. From this we derive existence theorems for two types of vectorial equilibrium problems, an analytic alternative and a minimax inequality involving three real functions.
Communications in Algebra, 2001
Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P n , n ≥ 3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p ≤ n − 1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P n , p ≤ n and the characteristic of the base field is zero. Finally, using our characterization, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu ([4]) on self-linking and complete intersections in positive characteristic.
Mathematische Zeitschrift, 2013
We address the conjecture of [Durfee1978], bounding the singularity genus pg by a multiple of the Milnor number µ for an n-dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely (n+1)!·pg ≤ µ, fails whenever the codimension r is greater than one. Moreover, we proposed a new inequality Cn,r · pg ≤ µ, and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a 'combinatorial inequality', that might have an independent interest.
Proceedings of the International Conference held in Trento, Italy, June 15 - 24, 1994, 1996
Proceedings of the American Mathematical Society, 1976
The concept of a generalized complete intersection (GCI) of affine schemes is introduced. The proofs of the following theorems are then
Inventiones mathematicae, 1991
Journal of Algebra, 2007
Let (S, n) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I * be the leading ideal of I in the associated graded ring gr n (S), and set R = S/I and m = n/I. In [GHK2], we prove that if µ G (I *) = n, then I * contains a homogeneous system {ξ i } 1≤i≤n of generators such that deg ξ i + 2 ≤ deg ξ i+1 for 2 ≤ i ≤ n−1, and htG(ξ1, ξ2, • • • , ξn−1) = 1, and we describe precisely the Hilbert series H(gr m (R), λ) in terms of the degrees c i of the ξ i and the integers d i , where di is the degree of Di = GCD(ξ1,. .. , ξi). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c 1 + 1, an ascending sequence of positive integers (c 1 , c 2 ,. .. , c n), and a descending sequence of integers (d1 = c1, d2,. .. , dn = 0) such that ci+1 − ci > di−1 − di > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.
Proceedings of the American Mathematical Society, 1998
Let k k be a field of characteristic 0. If I ⊂ k [ x , y , z ] I \subset k[x,y,z] is a complete intersection generated by three homogeneous elements of degrees d 1 , d 2 , d 3 d_1,d_2,d_3 with 2 ≤ d 1 ≤ d 2 ≤ d 3 2 \le d_1 \le d_2 \le d_3 , then the reduction of I I by a general linear form is minimally generated by three elements if and only if d 3 ≤ d 1 + d 2 − 2 d_3 \le d_1+d_2-2 .
Canadian Mathematical Bulletin, 1991
We deal here with the existence of half-way nonzero divisors for complete intersections and with related properties of their Hilbert functions.
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