93 research outputs found
On non Fundamental Group Equivalent Surfaces
In this paper we present an example of two polarized K3 surfaces which are
not Fundamental Group Equivalent (their fundamental groups of the complement of
the branch curves are not isomorphic; denoted by FGE) but the fundamental
groups of their related Galois covers are isomorphic. For each surface, we
consider a generic projection to CP^2 and a degenerations of the surface into a
union of planes - the "pillow" degeneration for the non-prime surface and the
"magician" degeneration for the prime surface. We compute the Braid Monodromy
Factorization (BMF) of the branch curve of each projected surface, using the
related degenerations. By these factorizations, we compute the above
fundamental groups. It is known that the two surfaces are not in the same
component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove
that furthermore they are not FGE equivalent, and thus they are not of the same
Braid Monodromy Type (BMT) (which implies that they are not a projective
deformation of each othe
The fundamental group of a Galois cover of CP^1 X T
Let T be the complex projective torus, and X the surface CP^1 X T. Let X_Gal
be its Galois cover with respect to a generic projection to CP^2. In this paper
we compute the fundamental group of X_Gal, using the degeneration and
regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and
group calculations. We show that pi_1(X_Gal) = Z^10.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-20.abs.htm
Moduli spaces and braid monodromy types of bidouble covers of the quadric
Bidouble covers of the quadric Q are parametrized by
connected families depending on four positive integers a,b,c,d. In the special
case where b=d we call them abc-surfaces.
Such a Galois covering admits a small perturbation yielding a general
4-tuple covering of Q with branch curve \De, and a natural Lefschetz
fibration obtained from a small perturbation of the composition of with
the first projection.
We prove a more general result implying that the braid monodromy
factorization corresponding to \De determines the three integers a,b,c in the
case of abc-surfaces. We introduce a new method in order to distinguish
factorizations which are not stably equivalent. This result is in sharp
contrast with a previous result of the first and third author, showing that the
mapping class group factorizations corresponding to the respective natural
Lefschetz pencils are equivalent for abc-surfaces with the same values of a+c,
b. This result hints at the possibility that abc-surfaces with fixed values of
a+c, b, although diffeomorphic but not deformation equivalent, might be not
canonically symplectomorphic.Comment: 38 pages, showkeys command cancelled with
On fundamental groups related to the Hirzebruch surface F_1
Given a projective surface and a generic projection to the plane, the braid
monodromy factorization (and thus, the braid monodromy type) of the complement
of its branch curve is one of the most important topological invariants, stable
on deformations. From this factorization, one can compute the fundamental group
of the complement of the branch curve, either in C^2 or in CP^2. In this
article, we show that these groups, for the Hirzebruch surface F_{1,(a,b)}, are
almost-solvable. That is - they are an extension of a solvable group, which
strengthen the conjecture on degeneratable surfaces.Comment: accepted for publication at "Sci. in China, ser. Math"; 22 pages, 11
figure
More Cappell-Shaneson spheres are standard
Akbulut has recently shown that an infinite family of Cappell-Shaneson
homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present
paper, a strictly larger family is shown to be standard by a simpler method.
This new approach uses no Kirby calculus except through the relatively simple
1979 paper of Akbulut and Kirby showing that the simplest example with
untwisted framing is standard. Instead, hidden symmetries of the original
Cappell-Shaneson construction are exploited. In the course of the proof, we
give an example showing that Gluck twists can sometimes be undone using
symmetries of fishtail neighborhoods.Comment: 11 pages, 2 figures. This (v2) is essentially the published version,
with minor mathematical improvements over v
Chart description for genus-two Lefschetz fibrations and a theorem on their stabilization
Chart descriptions are a graphic method to describe monodromy representations
of various topological objects. Here we introduce a chart description for
genus-two Lefschetz fibrations, and show that any genus-two Lefschetz fibration
can be stabilized by fiber-sum with certain basic Lefschetz fibrations.Comment: 17 pages, 18 figure
A stable classification of Lefschetz fibrations
We study the classification of Lefschetz fibrations up to stabilization by
fiber sum operations. We show that for each genus there is a `universal'
fibration f^0_g with the property that, if two Lefschetz fibrations over S^2
have the same Euler-Poincare characteristic and signature, the same numbers of
reducible singular fibers of each type, and admit sections with the same
self-intersection, then after repeatedly fiber summing with f^0_g they become
isomorphic. As a consequence, any two compact integral symplectic 4-manifolds
with the same values of (c_1^2, c_2, c_1.[w], [w]^2) become symplectomorphic
after blowups and symplectic sums with f^0_g.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper6.abs.htm
Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation
Let C be a locally planar curve. Its versal deformation admits a
stratification by the genera of the fibres. The strata are singular; we show
that their multiplicities at the central point are determined by the Euler
numbers of the Hilbert schemes of points on C. These Euler numbers have made
two prior appearances. First, in certain simple cases, they control the
contribution of C to the Pandharipande-Thomas curve counting invariants of
three-folds. In this context, our result identifies the strata multiplicities
as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C
is smooth away from a unique singular point, a special case of a conjecture of
Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with
the "U(infinity)" invariant of the link of the singularity. We make contact
with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.Comment: 16 page
Products of pairs of Dehn twists and maximal real Lefschetz fibrations
We address the problem of existence and uniqueness of a factorization of a
given element of the modular group into a product of two Dehn twists. As a
geometric application, we conclude that any maximal real elliptic Lefschetz
fibration is algebraic.Comment: Final version accepted for publicatio
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