85 research outputs found
Morsifications and mutations
We describe and investigate a connection between the topology of isolated
singularities of plane curves and the mutation equivalence, in the sense of
cluster algebra theory, of the quivers associated with their morsifications.Comment: Major revision and expansion. Several new results. 80 pages, 66
figure
More about vanishing cycles and mutation
The paper continues the discussion of symplectic aspects of Picard-Lefschetz
theory begun in "Vanishing cycles and mutation" (this archive). There we
explained how to associate to a suitable fibration over a two-dimensional disc
a triangulated category, the "derived directed Fukaya category" which describes
the structure of the vanishing cycles. The present second part serves two
purposes. Firstly, it contains various kinds of algebro-geometric examples,
including the "mirror manifold" of the projective plane. Secondly there is a
(largely conjectural) discussion of more advanced topics, such as (i)
Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse
theory, (iii) a proposed "dimensional reduction" algorithm for doing certain
Floer cohomology computations.Comment: 33 pages, LaTeX2e, 9 eps figure
Singular curves and quasi-hereditary algebras
In this article we construct a categorical resolution of singularities of an
excellent reduced curve , introducing a certain sheaf of orders on . This
categorical resolution is shown to be a recollement of the derived category of
coherent sheaves on the normalization of and the derived category of finite
length modules over a certain artinian quasi-hereditary ring depending
purely on the local singularity types of .
Using this technique, we prove several statements on the Rouquier dimension
of the derived category of coherent sheaves on . Moreover, in the case
is rational and projective we construct a finite dimensional quasi-hereditary
algebra such that the triangulated category of perfect complexes on
embeds into as a full subcategory.Comment: minor changes; to appear in IMR
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