8,528 research outputs found
Invariants of plane curve singularities and Pl\"ucker formulas in positive characteristic
We study classical invariants for plane curve singularities ,
an algebraically closed field of characteristic : Milnor number,
delta invariant, kappa invariant and multiplicity. It is known, in
characteristic zero, that and that
. For arbitrary characteristic,
Deligne prove that there is always the inequality by showing that
measures the wild vanishing cycles. By introducing new invariants
, we prove in this note that with equalities
if and only if the characteristic does not divide the multiplicity of any
branch of . As an application we show that if is "big" for (in fact
), then has no wild vanishing cycle. Moreover we obtain some
Pl\"ucker formulas for projective plane curves in positive characteristic.Comment: 15 pages; final version; to appear in the Annales de l'Institut
Fourie
Volume form on moduli spaces of d-differentials
Given , , and an integral vector
such that and , let
denote the moduli space of meromorphic
-differentials on Riemann surfaces of genus whose zeros and poles have
orders prescribed by . We show that
carries a canonical volume form that is parallel with respect to its affine
complex manifold structure, and that the total volume of
with respect to the measure induced by this volume form is finite.Comment: Streamlined, minor corrections added, definition of the volume form
independent of the choice of a d-th root of unit
The right classification of univariate power series in positive characteristic
While the classification of univariate power series up to coordinate change
is trivial in characteristic 0, this classification is very different in
positive characteristic. In this note we give a complete classification of
univariate power series , where is an algebraically closed
field of characteristic by explicit normal forms. We show that the right
determinacy of is completely determined by its support. Moreover we prove
that the right modality of is equal to the integer part of , where
is the Milnor number of . As a consequence we prove in this case that
the modality is equal to the proper modality, which is the dimension of the
-constant stratum in an algebraic representative of the semiuniversal
deformation with trivial section.Comment: 17 pages, final versio
Translation surfaces and the curve graph in genus two
Let be a (topological) compact closed surface of genus two. We associate
to each translation surface a subgraph
of the curve graph of . The vertices of this subgraph are free homotopy
classes of curves which can be represented either by a simple closed geodesic,
or by a concatenation of two parallel saddle connections (satisfying some
additional properties) on . The subgraph is by
definition -invariant. Hence, it may be seen as
the image of the corresponding Teichm\"uller disk in the curve graph. We will
show that is always connected and has infinite
diameter. The group of affine automorphisms of
preserves naturally , we show that
is precisely the stabilizer of in . We also prove that is
Gromov-hyperbolic if is completely periodic in the sense of Calta.
It turns out that the quotient of by is closely related to McMullen's prototypes in the case
is a Veech surface in . We finally show that this
quotient graph has finitely many vertices if and only if is a
Veech surface for in both strata and
.Comment: 47 pages, 17 figures. Minor changes, some proofs improved. Comments
welcome
Right unimodal and bimodal singularities in positive characteristic
The problem of classification of real and complex singularities was initiated
by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t.
right equivalence. The classification of right simple singularities in positive
characteristic was achieved by Greuel and the author in 2014. In the present
paper we classify right unimodal and bimodal singularities in positive
characteristic by giving explicit normal forms. Moreover we completely
determine all possible adjacencies of simple, unimodal and bimodal
singularities. As an application we prove that, for singularities of right
modality at most 2, the -constant stratum is smooth and its dimension is
equal to the right modality. In contrast to the complex analytic case, there
are, for any positive characteristic, only finitely many 1-dimensional (resp.
2-dimensional) families of right class of unimodal (resp. bimodal)
singularities. We show that for fixed characteristic of the ground field,
the Milnor number of satisfies , if the right modality of
is at most 2.Comment: 19 page
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