The equations of almost complete intersections

In this paper we inject four Hilbert functions in the determination of the defining equations of the Rees algebra of almost complete intersections of finite co-length. Because three of the corresponding modules are Artinian, some of these relationships are very effective, with the novel approach opening up tracks to the determination of the equations and also to processes of going from homologically defined sets of equations to higher degrees ones. While not specifically directed towards the extraction of elimination equations, it will show how some of these arise naturally.     

On Landsberg's criterion for complete intersections

Partially supported by ISF grant MSC000     

Topology of complete intersections

Let X _n (d) and X _n (d') be two n-dimensional complete intersections with the same total degree d. In this paper we prove that, if n is even and d has no prime factors less than , then X _n (d) and X _n (d') are homotopy equivalent if and only if they have the same Euler characteristics and signatures. This confirms a conjecture of Libgober and Wood [16]. Furthermore, we prove that, if d has no prime factors less than , then X _n (d) and X _n (d') are homeomorphic if and only if their Pontryagin classes and Euler characteristics agree. Received: September 6, 1996     

On almost complete intersections

Let R be a local ring such that R=S/I where S is a regular local ring and I is a prime ideal of height r. In this paper it is shown that if I is minimally generated by r+1 elements, then there exists an R-homomorphism : K_RR^r+1 such that is an injection and R^r+1/(K_R)I/I² where K_R:=Ext _S ^r (R,S) the canonical module of R. Moreover, in case where S is a locality over a perfect field k, it is also shown that if R is Cohen-Macaulay and I² is a primary ideal, then the homological dimension of the differential module _R/k is infinite.The author wishes to thank his colleague Mr.Y.Aoyama for valuable discussions in connection with this subject.     

Birational rigidity of complete intersections

We prove that every smooth complete intersection \(X=X_{d_{1}, \ldots , d_{s}}\subset \mathbb {P}^{\sum _{i=1}^{s}d_{i}}\) defined by s hypersurfaces of degree \(d_{1}, \ldots , d_{s}\) is birationally superrigid if \(5s +1\le \frac{2(\sum _{i=1}^{s}d_{i}+1)}{\sqrt{\prod _{i=1}^{s}d_{i}}}\). In particular, X is non-rational and \({{\mathrm{Bir}}}(X)={{\mathrm{Aut}}}(X)\). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.     

Ext-symmetry over quantum complete intersections

We show that symmetry in the vanishing of cohomology holds for graded modules over quantum complete intersections. Moreover, symmetry holds for all modules if the algebra is symmetric. Received: 1 December 2008     

Monomial conjecture and complete intersections

The Monomial Conjecture is an assertion about all systems of parameters in an arbitrary noetherian local ringA. We reduce this to a statement about a closely related complete intersection ringR of the same dimension. This statement again involves parameters, but in an ostensibly different way, raising new questions.     

The n-lie property of the Jacobian as a condition for complete integrability

We prove that an associative commutative algebra U with derivations D ₁, ..., D _n ε DerU is an n-Lie algebra with respect to the n-multiplication D ₁ ^ ? ^ D _n if the system {D ₁, ..., D _n } is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of n-Lie multiplications. A differential system {D ₁, ..., D _n } of rank n on a manifold M ^m is in involution if and only if the space of smooth functions on M is an n-Lie algebra with respect to the Jacobian Det(D _i u _j ).