Dan Laksov

Weierstrass points and gap sequences for families of curves

The theory of Weierstrass points and gap sequences for linear series on smooth curves is generalized to smooth families of curves with geometrically irreducible fibers, and over an arbitrary base scheme.

These are the differentials of order

We answer P.-A. Meyer’s question “Qu’est ce qu’une différentielle d’ordre n?”. In fact, we present a general theory of higher order differentials based upon a construction of universal objects for higher order differentials. Applied to successive tangent spaces on a differentiable manifold, our theory gives the higher order differentials of Meyer as well as several new results on differentials on differentiable manifolds. In addition our approach gives a natural explanation of the quite mysterious multiplicative structure on higher order differentials observed by Meyer. Applied to iterations of the first order Kähler differentials our theory gives an algebra of higher order differentials for any smooth scheme. We also observe that much of the recent work on higher order osculation spaces of varieties fits well into the framework of our theory.

On Zanello’s lower bound for level algebras

We consider the proof of Soderberg of Zanellos lower bound for the Hilbert function of level algebras from the point of view of vector spaces. Our results, when specialised to level algebras, generalise those of Zanello and Soderberg to the case when the modules involved may have nontrivial annihilators. In the process we clarify why the methods of Zanello and Soderberg consist of two distinct parts. As a contrast we show that for polynomial rings, Zanellos bound, in the generic case, can be obtained by simple manipulations of numbers without dividing into two separate cases. We also consider the inclusion-exclusion principle of dimensions of vector spaces used by Zanello in special cases. It turns out that the resulting alternating sums are extremely difficult to handle and have many unexpected properties. This we illustrate by a couple of results and examples. The examples show that the inclusion-exclusion principle does not hold for vector spaces in the way it is used by Zanello.

A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus

In previous work we developed a general formalism for equivariant Schubert calculus of grassmannians consisting of a basis theorem, a Pieri formula and a Giambelli formula. Part of the work consists in interpreting the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians. Here we present an extract of the theory containing the essential features of this ring. In particular we emphasize the importance of the GKM condition. Our formalism and methods are influenced by the combinatorial formalism given by A. Knutson and T. Tao for equivariant cohomology of grassmannians, and of the use of factorial Schur polynomials in the work of L.C. Mihalcea.