2,273 research outputs found
Higher discriminants and the topology of algebraic maps
We show that the way in which Betti cohomology varies in a proper family of
complex algebraic varieties is controlled by certain "higher discriminants" in
the base. These discriminants are defined in terms of transversality
conditions, which in the case of a morphism between smooth varieties can be
checked by a tangent space calculation. They control the variation of
cohomology in the following two senses: (1) the support of any summand of the
pushforward of the IC sheaf along a projective map is a component of a higher
discriminant, and (2) any component of the characteristic cycle of the proper
pushforward of the constant function is a conormal variety to a component of a
higher discriminant.
The same would hold for the Whitney stratification of the family, but there
are vastly fewer higher discriminants than Whitney strata. For example, in the
case of the Hitchin fibration, the stratification by higher discriminants gives
exactly the {\delta} stratification introduced by Ngo.Comment: v2: proofs rewritten in the language of microsupport, and added
example of integrable system
On Finite Order Invariants of Triple Points Free Plane Curves
We describe some regular techniques of calculating finite degree invariants
of triple points free smooth plane curves . They are a direct
analog of similar techniques for knot invariants and are based on the calculus
of {\em triangular diagrams} and {\em connected hypergraphs} in the same way as
the calculation of knot invariants is based on the study of chord diagrams and
connected graphs.
E.g., the simplest such invariant is of degree 4 and corresponds to the
diagram consisting of two triangles with alternating vertices in a circle in
the same way as the simplest knot invariant (of degree 2) corresponds to the
2-chord diagram . Also, following V.I.Arnold and other authors we
consider invariants of {\em immersed} triple points free curves and describe
similar techniques also for this problem, and, more generally, for the
calculation of homology groups of the space of immersed plane curves without
points of multiplicity for any $k \ge 3.
A support theorem for nested Hilbert schemes of planar curves
Consider a family of integral complex locally planar curves. We show that
under some assumptions on the basis, the relative nested Hilbert scheme is
smooth. In this case, the decomposition theorem of Beilinson, Bernstein and
Deligne asserts that the pushforward of the constant sheaf on the relative
nested Hilbert scheme splits as a direct sum of shifted semisimple perverse
sheaves. We will show that no summand is supported in positive codimension
Low-dimensional Singularities with Free Divisors as Discriminants
We present versal complex analytic families, over a smooth base and of fibre
dimension zero, one, or two, where the discriminant constitutes a free divisor.
These families include finite flat maps, versal deformations of reduced curve
singularities, and versal deformations of Gorenstein surface singularities in
C^5. It is shown that such free divisors often admit a "fast normalization",
obtained by a single application of the Grauert-Remmert normalization
algorithm. For a particular Gorenstein surface singularity in C^5, namely the
simple elliptic singularity of type \tilde A_4, we exhibit an explicit
discriminant matrix and show that the slice of the discriminant for a fixed
j-invariant is the cone over the dual variety of an elliptic curve.Comment: 29 pages, misprints and references correcte
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