658 research outputs found
Tropical Discriminants
Tropical geometry is used to develop a new approach to the theory of
discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky.
The tropical A-discriminant, which is the tropicalization of the dual variety
of the projective toric variety given by an integer matrix A, is shown to
coincide with the Minkowski sum of the row space of A and of the
tropicalization of the kernel of A. This leads to an explicit positive formula
for the extreme monomials of any A-discriminant, without any smoothness
assumption.Comment: Major revisions, including several improvements and the correction of
Section 5. To appear: Journal of the American Mathematical Societ
The supersingular locus of the Shimura variety for GU(1,n-1) over a ramified prime
We analyze the geometry of the supersingular locus of the reduction modulo p
of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q,
in the case that p is ramified. We define a stratification of this locus and
show that its incidence complex is closely related to a certain Bruhat-Tits
simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety
associated to some symplectic group over F_p and some Coxeter element. The
closure of each stratum is a normal projective variety with at most isolated
singularities. The results are analogous to those of Vollaard/Wedhorn in the
case when p is inert.Comment: A few more corrections, to appear in Math. Zeitschrif
Modular forms and elliptic curves over the cubic field of discriminant -23
Let F be the cubic field of discriminant -23 and let O be its ring of
integers. By explicitly computing cohomology of congruence subgroups of
GL(2,O), we computationally investigate modularity of elliptic curves over F.Comment: Incorporated referee's comment
Semi-inverted linear spaces and an analogue of the broken circuit complex
The image of a linear space under inversion of some coordinates is an affine
variety whose structure is governed by an underlying hyperplane arrangement. In
this paper, we generalize work by Proudfoot and Speyer to show that circuit
polynomials form a universal Groebner basis for the ideal of polynomials
vanishing on this variety. The proof relies on degenerations to the
Stanley-Reisner ideal of a simplicial complex determined by the underlying
matroid. If the linear space is real, then the semi-inverted linear space is
also an example of a hyperbolic variety, meaning that all of its intersection
points with a large family of linear spaces are real.Comment: 16 pages, 1 figure, minor revisions and added connections to the
external activity complex of a matroi
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