arXiv subjects
Publications and source records attributed to Anand Patel.
We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.
Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory, and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.
We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties. Going further, we study the map that associates to a linear projection its ramification divisor. We show that this map is dominant for most (but not all!) varieties of minimal degree, using (linked) limit linear series of higher rank. We find the degree of this map in some cases, extending the classical appearance of Catalan numbers in the geometry of rational normal curves, and give a geometric explanation of its fibers in terms of torsion points of naturally occurring elliptic curves in the case of the Veronese surface and the quartic rational surface scroll.
We describe a sequence of effective divisors on the Hurwitz space $H_{d,g}$ for $d$ dividing $g-1$ and compute their cycle classes on a partial compactification. These divisors arise from vector bundles of syzygies canonically associated to a branched cover. We find that the cycle classes are all proportional to each other.
We investigate the global variation of moduli of linear sections of a general hypersurface. We prove a "generic Torelli" result for a large proportion of cases, and we obtain a complete picture of the global variation of moduli of line slices of a general hypersurface.
This is an expanded version of the two papers "Interpolation of Varieties of Minimal Degree" and "Interpolation Problems: Del Pezzo Surfaces." It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. More recently, it was shown that one can always find nonspecial curves through the expected number of general points and linear spaces. After some expository material regarding scrolls, we consider the generalization of this question to varieties of all dimensions and explain why smooth varieties of minimal degree satisfy interpolation. We give twenty-two equivalent formulations of interpolation. We also classify when Castelnuovo curves satisfy weak interpolation. In the appendix, we prove that del Pezzo surfaces satisfy weak interpolation. Our techniques for proving interpolation include deformation theory, degeneration and specialization, and association.
We investigate two families of divisors which we expect to play a distinguished role in the global geometry of Hurwitz space. In particular, we show that they are extremal and rigid in the small degree regime $d \leq 5$. We further show their significance in the problem of computing the sweeping slope of Hurwitz space in these degrees. In the process, we prove various general results about the divisor theory of Hurwitz space, including a proof of the independence of the boundary components of the admissible covers compactification. Some basic open questions and further directions are discussed at the end.
We investigate the resolution of a general branched cover $α: C \to {\bf P}^1$ in its relative canonical embedding $C \subset {\bf P} E$. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati-Ekedahl bundle, or "bundle of quadrics" $F$ - the first bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies $g = 1 \mod d$.
We show that certain divisors of Brill-Noether and Gieseker-Petri type span extremal rays of the effective cone in the moduli space of stable genus one curves with $n$ ordered marked points. In particular, they are different from the infinitely many extremal rays found in \cite{ChenCoskun}.
We prove that the rational Picard group of the simple Hurwitz space ${\mathcal H}_{d,g}$ is trivial for $d$ up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of nodal curves on Hirzebruch surfaces.
Comparing strains within the same microbial species has proven effective in the identification of genes and genomic regions responsible for virulence, as well as in the diagnosis and treatment of infectious diseases. In this paper, we present Sibelia, a tool for finding synteny blocks in multiple closely related microbial genomes using iterative de Bruijn graphs. Unlike most other tools, Sibelia can find synteny blocks that are repeated within genomes as well as blocks shared by multiple genomes. It represents synteny blocks in a hierarchy structure with multiple layers, each of which representing a different granularity level. Sibelia has been designed to work efficiently with a large number of microbial genomes; it finds synteny blocks in 31 S. aureus genomes within 31 minutes and in 59 E.coli genomes within 107 minutes on a standard desktop. Sibelia software is distributed under the GNU GPL v2 license and is available at: https://github.com/bioinf/Sibelia Sibelia's web-server is available at: http://etool.me/software/sibelia
We establish sharp bounds for the slopes of curves in $\bar{M}_g$ that sweep the locus of trigonal curves, proving Stankova-Frenkel's conjectured bound of $7+6/g$ for even $g$ and obtaining the bound $7+20/(3g+1)$ for odd $g$. For even $g$, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd $g$, we describe the analogous extremal effective divisor and give a similar explicit expression.