Algebraic Geometry

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians G(k,n) to some Gorenstein toric Fano varieties P(k,n) with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections X⊂G(k,n) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.

We added an additional result (theorem 1.6) that strengthenns our main theorem in the G=GL-case by establishing an equivalence of tensor categories.

In this paper we reduce the problem of counting the number of connected components in the intersection of two opposite open Schubert cells in the variety of real complete flags to a purely combinatorial question of counting the number of orbits of a certain intriguing group action in the space of upper triangular matrices with {0,1}-valued entries. The crucial step of our reduction uses the parametrization of the space of real unipotent totally positive upper triangular matrices introduced by Lusztig and Berenstein, Fomin, Zelevinski.

Suppose A is an abelian variety over a field F , and ℓ is a prime not equal to the characteristic of F . Let F Φ,ℓ (A) denote the smallest extension of F such that the Zariski closure of the image of the ℓ -adic representation associated to A is connected. Serre introduced this field, and proved that when F is a finitely generated extension of Q , F Φ,ℓ (A) does not depend on the choice of ℓ . In this paper we study extensions F Φ,ℓ (B)/F for twists B of a given abelian variety, especially when the abelian varieties are of Weil type.

Let A be a Noetherian local domain, N be a finitely generated torsion- free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C:=Proj(A[M]) and r:=dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]_p is not a finitely generated module over A[M]_p, and denote the preimage of W in C by E. We prove this: (1) dim E=r-1 if either (a) N is free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex- analytic variety, and applied it to perfect a characterization of Thom's A_f-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of B"oger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by the second author, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M].

In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of class field theory.

We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the ample cone gives rise to an algorithm for constructing new ample divisors. We then propose a conjecture relating continued fractions approximations and Seshadri-like constants of line bundles over rational surfaces. By applying our algorithm recursively we verify our conjecture in many cases and obtain new asymptotic estimates on these constants. Finally, we explain the intuition behind the gluing theorem in terms of symplectic geometry and propose generalizations.

Let A be a local noetherian ring and N be a locally sheaf on the projective space P 3 A : one proves easily that there exists a family C of (smooth connected) curves contained in P 3 A , flat over A , and an integer h such that the ideal sheaf J of C has a resolution 0→P→N→J→0 where P is a direct sum of invertible sheaves O P (− n i ) . In this paper we determine, for a given sheaf N , all the families of curves with such a resolution, especially the minimal ones (corresponding to the minimum value of h ). It gives a description of the biliaison class related to N , and a tool for constructing families of space curves.

We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology encodes the tangency characteristic numbers.

We consider the cones of curves and divisors on the moduli space of stable pointed rational curves,M_n, and on the quotient by the symmetric group, Q_n, which is a moduli space of pairs. We find generators for contractible extremal rays of the cone of curves NE_1(M_n), and for the cone of divisors NE^1(Q_n). This second cone turns out to be simplicial. We give complete descriptions of NE_1(M_n) and NE_1(Q_n) for small n (< 8 in the first case, < 11 in the second). We also have results of independent interest on when curves in a divisor generate the cone of curves of the ambient variety.