Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r =
  f(x)$. II   Previous work by the authors (this journal, \vol{60} (2008), 1009-1044)
produced equations that hold on certain loci of the Jacobian of a cyclic
$C_{rs}$ curve. A curve of this type generalizes elliptic curves, and the
equations in question are given in terms of (Klein's) generalization of
Weierstrass' $\sigma$-function. The key tool is a matrix with entries that are
polynomial in the coordinates of the affine plane model of the curve, thus can
be expressed in terms of $\sigma$ and its derivatives. The key geometric loci
on the Jacobian of the curve give a stratification of Brill-Noether type. The
results are of the type of Riemann-Kempf singularity theorem, the methods are
germane to those used by J.D. Fay, who gave vanishing tables for Riemann's
$\theta$-function and its derivatives. The main objects we use were developed
by several contemporary authors, aside from the classical definitions:
meromorphic differentials were expressed in terms of the coordinates mainly by
V.M. Buchstaber, J.C. Eilbeck, V.Z. Enolski, D.V. Leykin, and Taylor expansions
for $\sigma$ in terms of Schur polynomials also contributed by A. Nakayashiki,
in terms of Sato's $\tau$-function. Within this framework, following specific
results for $\sigma$-derivatives given by Y. \^Onishi, we arrive at our main
results, namely statements on the vanishing on given strata of the partial
derivatives of $\sigma$ indexed by Young-diagrams subsets that can be worked
out in terms of the Weierstrass semigroup of the curve at its point at
infinity. The combinatorial statements hold not only for Jacobians but for the
stratification of Sato's infinite-dimensional Grassmann manifold as well.

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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph