arXiv subjects

Aaron Bertram

Publications and source records attributed to Aaron Bertram.

24 records · Page 2Linked to original sources

arXiv1997

An Application of a Log Version of the Kodaira Vanishing Theorem to Embedded Projective Varieties

Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain twists of powers of the ideal sheaf of Y. This is applied to get rather non-trivial vanishing in case Y is a universal determinantal variety (i.e. quadratic Veronese, Segre embedding or Grassmannian of P^1's) or a curve embedded by a complete linear series of high degree. The key in the determinantal cases is to use the theory of complete objects. While the theories of complete quadrics and linear maps yield the desired results, a corresponding theory of complete skew forms seems not have been previously worked out. The desired results are obtained in an appendix.

alg-geom↗

arXiv1997

Stable pairs and log flips

This paper has two parts. In the first part, we review stable pairs and triples on curves, leading up to Thaddeus' diagram of flips and contractions starting from the blow-up of projective space along a curve embedded by a complete linear series of the form K + ample. In the second part, we identify log canonical divisors which exhibit Thaddeus' flips and contractions as "log" flips and contractions in the sense of the log-minimal-model program.

alg-geom↗

arXiv1997

Quantum Schubert Calculus

This paper works out the versions of the classical Giambelli and Pieri formulas in the context of quantum cohomology of a complex Grassmannian.

alg-geom↗

arXiv1997

Quantum multiplication of Schur polynomials

We give formulas for the products of classes of Schubert varieties in the quantum cohomology rings of Grassmannians, in terms of the combinatorics of partitions and tableaux.

alg-geom↗

arXiv1994

Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian

The intuitive notion of the Gromov invariant for maps from a Riemann surface to a Grassmannian is shown to agree with the definition in \cite{BDW}. Also, an induction on the genus is proved, which extends the results of \cite{BDW} to a computation of all Gromov invariants associated to G(2,k). This is shown to agree with the conjectured formula of Vafa and Intriligator.

alg-geom↗

arXiv1993

Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective scheme and is dominated by the algebraic compactification arising as a Grothendieck Quot scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as Gromov invariants) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.

alg-geom↗