Brian Osserman

Rational functions with given ramification in characteristic p

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of rational functions (up to automorphism of the target) on the projective line with ramification to order e i at general points P i , in the case that all e i are less than the characteristic. Unlike the case of characteristic 0, the answer is not given by Schubert calculus, nor is the number of maps always finite for distinct P i , even in the tamely ramified case. However, finiteness for general P i , obtained by exploiting the relationship to branched covers, is a key part of the argument.

BRILL-NOETHER LOCI WITH FIXED DETERMINANT IN RANK 2

In the 1990s, Bertram, Feinberg and Mukai examined Brill-Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat a much broader range of fixed-determinant Brill-Noether loci. The main technique is a careful study of symplectic Grassmannians and related concepts.

SPECIAL DETERMINANTS IN HIGHER-RANK BRILL–NOETHER THEORY

Extending our previous study of modified expected dimensions for rank-2 Brill–Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmannians, which are the loci inside Grassmannians corresponding to subspaces which are simultaneously isotropic for a family of multilinear alternating forms on the ambient vector space. In the case of rank 2 with two-dimensional spaces of sections, we adapt arguments due to Teixidor i Bigas to show that our new modified expected dimensions are in fact sharp.

The number of linear series on curves with given ramification

We use Eisenbud and Harris’ theory of limit linear series (1986) to show that for a general smooth curve of genus g in characteristic 0, with general points Pi and indices ei such that P i(ei − 1) = 2d − 2− g, G 1 d (C, {(Pi, ei)}i) is made up of reduced points. We give a formula for the number of points, showing that it agrees with various known special cases. We also conjecture a corresponding reducedness result and formula for g d s of any dimension, and reduce this to the case of three points on P, where one need no longer consider moduli or generality.

Frobenius-unstable bundles and

We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for p-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve, and applying the results of additional works by the author. We also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different p-rank.

p

In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on the existence and non-existence of tamely branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy cycles, and the first class of results are obtained by intrinsically algebraic and positive-characteristic arguments.

-curvature

In [7], a new construction of limit linear series is presented which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The boundary of the compactification consists of crude limit series, and maps with positivedimensional fibers to crude limit series of Eisenbud and Harris. In this paper, we carry out a careful analysis of the linked Grassmannian to obtain an upper bound on the dimension of the fibers of the map on crude limit series, thereby concluding an upper bound on the dimension of the locus of crude limit series, and obtaining a simple proof of the Brill-Noether theorem using only the limit linear series machinery. We also see that on a general reducible curve, even crude limit series may be smoothed to nearby fibers.

Linear series and the existence of branched covers

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.

Linked Grassmannians and crude limit linear series

We observe that if we are interested primarily in degeneration arguments, there is a weaker notion of (semi)stability for vector bundles on reducible curves, which is sufficient for many applications, and does not depend on a choice of polarization. We introduce and explore the basic properties of this alternate notion of (semi)stability. In a complementary direction, we record a proof of the existence of semistable extensions of vector bundles in suitable degenerations.

Flatness of the linked Grassmannian

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve C of genus g to P 1 with prescribed ramification also yields weaker results when working over the real numbers or p-adic fields. Specifically, let k be such a field: we see that given g, d, n, and e 1 ,..e n satisfying Σ ι (e i -1) = 2d - 2 - g, there exists smooth curves C of genus g together with points P 1 ,..., P n such that all maps from C to P 1 can, up to automorphism of the image, be defined over k. We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case C = P 1 , n = 3, and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.