186 research outputs found
Schubert decompositions for quiver Grassmannians of tree modules
Let be a quiver, a representation of with an ordered basis \cB
and \ue a dimension vector for . In this note we extend the methods of
\cite{L12} to establish Schubert decompositions of quiver Grassmannians
\Gr_\ue(M) into affine spaces to the ramified case, i.e.\ the canonical
morphism from the coefficient quiver of w.r.t.\ \cB is not
necessarily unramified.
In particular, we determine the Euler characteristic of \Gr_\ue(M) as the
number of \emph{extremal successor closed subsets of }, which extends the
results of Cerulli Irelli (\cite{Cerulli11}) and Haupt (\cite{Haupt12}) (under
certain additional assumptions on \cB).Comment: 22 page
Scheme theoretic tropicalization
In this paper, we introduce ordered blueprints and ordered blue schemes,
which serve as a common language for the different approaches to
tropicalizations and which enhances tropical varieties with a schematic
structure. As an abstract concept, we consider a tropicalization as a moduli
problem about extensions of a given valuation between ordered
blueprints and . If is idempotent, then we show that a
generalization of the Giansiracusa bend relation leads to a representing object
for the tropicalization, and that it has yet another interpretation in terms of
a base change along . We call such a representing object a scheme theoretic
tropicalization.
This theory recovers and improves other approaches to tropicalizations as we
explain with care in the second part of this text.
The Berkovich analytification and the Kajiwara-Payne tropicalization appear
as rational point sets of a scheme theoretic tropicalization. The same holds
true for its generalization by Foster and Ranganathan to higher rank
valuations.
The scheme theoretic Giansiracusa tropicalization can be recovered from the
scheme theoretic tropicalizations in our sense. We obtain an improvement due to
the resulting blueprint structure, which is sufficient to remember the
Maclagan-Rinc\'on weights.
The Macpherson analytification has an interpretation in terms of a scheme
theoretic tropicalization, and we give an alternative approach to Macpherson's
construction of tropicalizations.
The Thuillier analytification and the Ulirsch tropicalization are rational
point sets of a scheme theoretic tropicalization. Our approach yields a
generalization to any, possibly nontrivial, valuation with
idempotent and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the
paper, please cf. the paragraph "Differences to previous versions" in the
introductio
Automorphic forms for elliptic function fields
Let be the function field of an elliptic curve over \F_q. In this
paper, we calculate explicit formulas for unramified Hecke operators acting on
automorphic forms over . We determine these formulas in the language of the
graph of an Hecke operator, for which we use its interpretation in terms of
-bundles on . This allows a purely geometric approach, which involves,
amongst others, a classification of the -bundles on .
We apply the computed formulas to calculate the dimension of the space of
unramified cusp forms and the support of a cusp form. We show that a cuspidal
Hecke eigenform does not vanish in the trivial -bundle. Further, we
determine the space of unramified -toroidal automorphic forms where is
the quadratic constant field extension of . It does not contain non-trivial
cusp forms. An investigation of zeros of certain Hecke -series leads to the
conclusion that the space of unramified toroidal automorphic forms is spanned
by the Eisenstein series E(\blanc,s) where is a zero of the zeta
function of ---with one possible exception in the case that is even and
the class number equals .Comment: 26 page
The geometry of blueprints. Part I: Algebraic background and scheme theory
In this paper, we introduce the category of blueprints, which is a category
of algebraic objects that include both commutative (semi)rings and commutative
monoids. This generalization allows a simultaneous treatment of ideals resp.\
congruences for rings and monoids and leads to a common scheme theory. In
particular, it bridges the gap between usual schemes and -schemes
(after Kato, Deitmar and Connes-Consani). Beside this unification, the category
of blueprints contains new interesting objects as "improved" cyclotomic field
extensions of and "archimedean valuation
rings". It also yields a notion of semiring schemes.
This first paper lays the foundation for subsequent projects, which are
devoted to the following problems: Tits' idea of Chevalley groups over
, congruence schemes, sheaf cohomology, -theory and a unified
view on analytic geometry over , adic spaces (after Huber),
analytic spaces (after Berkovich) and tropical geometry.Comment: Slightly revised and extended version as in print. 51 page
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