Moduli spaces of rational curves on del Pezzo surfaces and Fano threefolds

I will talk about several results on the geometry of moduli spaces of rational curves on del Pezzo surfaces and Fano threefolds with regard to the Geometric Manin's Conjecture. The key question that I will discuss is: what can be said about the number of irreducible components of the moduli space as the anti-canonical degree of the curves increases? This is joint work with Brian Lehmann, Eric Riedl, and Sho Tanimoto.

Height pairings for algebraic cycles

There has been considerable work over the years on height pairings for algebraic cycles. Geometrically, height pairings for 0-cycles and divisors on curves and on abelian varieties are now quite well understood. There has even been considerable progress arithmetically on the Birch and Swinnerton Dyer conjecture. The problem for cycles of dimension > 0 is more difficult, though it would seem to be important. There is, for example, a good deal of recent work on relations with the Langlands program.

I will explain a few ideas and examples.

I. The basic geometric and motivic setup. The Selmer group, Birch and Swinnerton Dyer, Tamagawa numbers.
II ''Stupid heights''; functions on Hilbert schemes.
III. p-adic Hodge theory and local heights.
IV. The double point example; Beilinson's theorem.

The absolute Grothendieck conjecture is false for Fargues-Fontaine curves

I will explain the theorem in the title, and its relevance and sketch a proof and some explicit examples.

Quasi-unipotency of local monodromy

We consider some variations and applications of Grothendieck's theorem on local monodromy to finite abelian coverings of varieties.

Algebraization of point set topology

In a collaboration with Alok Maharana, we studied the work of Stone, Hochster, Isbell, Joyal, Johnstone and others on locales and sober topological spaces. As a result we can exhibit a locale or a sober topological space as a dense sublocale of an algebraic scheme over a field. Such dense sublocales are understood in terms of nuclei. The embedding is functorial.

Random thoughts and an old friend

Back in the previous century (1998-1999) I was a postdoc with Mohan Kumar (and, de facto, with Prabhakar Rao). It was an exciting time for me as these were two of my mathematical heroes. Inspired by Mohan's remarkable paper ''Construction of rank two vector bundles on P⁴ in positive characteristic'', we worked on a project which used the tautological bundle on the Grassmannian Gr(2,4) to give a very concrete construction of low rank vector bundles on P⁴. This was over an algebraically closed field with positive characteristic. Working in positive characteristic was new for me as I had always worked over C. The project definitely opened my mind to new ways of thinking. Today's talk revisits the tautological bundle on Gr(2,4) but over the reals using ''random algebraic geometry''. We consider the problem of determining the expected number of real lines lying on a random real cubic surface in P³.

Deformation of canonical maps and its applications

A framework was developed in a joint work with F. J. Gallego and M. Gonzalez to systematically deal with the deformation of finite morphisms, multiple scheme structures on algebraic varieties and their smoothing. There are several applications of this framework. In this talk I will talk about some of them. First is the description of some components of the moduli space of varieties of general type in all dimensions. In particular, we show the existence of components of the moduli space of general type in all dimensions, that are analogue of the moduli space of curves of genus g>2. Secondly, we give a new method to construct smooth varieties in projective space embedded by complete sub canonical linear series within the range of the Hartshorne conjecture and beyond. Are all of them complete intersections? We also construct systematically, smooth non complete intersection subvarieties embedded by complete linear series outside the range of the Hartshorne conjecture. As a byproduct, we construct simple canonical varieties of any dimension, expanding the original question posed by Enriques for algebraic surfaces. This is joint work with F. J. Gallego, J Mukherjee and D. Raychaudhury.

Vector bundles on varieties

I will revisit the important work of Professor Kumar on vector bundles on projective spaces, review some collaborations with Chris Peterson as well as with G. Ravindra on related topics and finish with some recent extensions of these ideas.

On finite presentation for the tame fundamental group

This is a report on joint work with H. Esnault and M. Schusterman.

Recall that the etale fundamental group of a variety over an algebraically closed field of characteristic 0 is known to be a finitely presented profinite group; this is proved by first reducing to varieties over the complex numbers, and then comparing with the topological fundamental group. In positive characteristics, even if we restrict to smooth varieties, finite generation fails in general for etale fundamental groups of non-proper varieties (eg, for the affine line).

For a smooth variety with a smooth, projective compactification with a SNC boundary divisor, we show that the tame fundamental group is a finitely presented profinite group. In particular, this holds for the fundamental groups of smooth projective varieties.

A survey of the illustrious Jacobian conjecture

The celebrated Jacobian Conjecture asserts:

Let F be a polynomial map from Cⁿ to Cⁿ. If the Jacobian determinant of F is everywhere non-vanishing, then F is a polynomial automorphism.

This conjecture, now 83 years old and still unsolved for n>1, can be viewed as a problem in differential geometry, algebraic geometry, commutative algebra, analysis, topology, and combinatorics. Over the years it has proved to be remarkably elusive and deceptive, with numerous false proofs appearing, some published by notable mathematicians. In this talk we will recount this sometimes amusing story, name the key players, discuss some of the problem’s subtle obstructions, state known reductions of the problem, identify cases where it has been proved to be true, and state some important equivalent formulations.