16,522 research outputs found
Minimality and irreducibility of symplectic four-manifolds
We prove that all minimal symplectic four-manifolds are essentially
irreducible. We also clarify the relationship between holomorphic and
symplectic minimality of K\"ahler surfaces. This leads to a new proof of the
deformation-invariance of holomorphic minimality for complex surfaces with even
first Betti number which are not Hirzebruch surfaces.Comment: final version; cosmetic changes only; to appear in International
Mathematics Research Notice
The implicit equation of a canal surface
A canal surface is an envelope of a one parameter family of spheres. In this
paper we present an efficient algorithm for computing the implicit equation of
a canal surface generated by a rational family of spheres. By using Laguerre
and Lie geometries, we relate the equation of the canal surface to the equation
of a dual variety of a certain curve in 5-dimensional projective space. We
define the \mu-basis for arbitrary dimension and give a simple algorithm for
its computation. This is then applied to the dual variety, which allows us to
deduce the implicit equations of the the dual variety, the canal surface and
any offset to the canal surface.Comment: 26 pages, to be published in Journal of Symbolic Computatio
The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14
The Abel-Jacobi maps of the families of elliptic quintics and rational
quartics lying on a smooth cubic threefold are studied. It is proved that their
generic fiber is the 5-dimensional projective space for quintics, and a smooth
3-dimensional variety birational to the cubic itself for quartics. The paper is
a continuation of the recent work of Markushevich-Tikhomirov, who showed that
the first Abel-Jacobi map factors through the moduli component of stable rank 2
vector bundles on the cubic threefold with Chern numbers
obtained by Serre's construction from elliptic quintics, and that the
factorizing map from the moduli space to the intermediate Jacobian is \'etale.
The above result implies that the degree of the \'etale map is 1, hence the
moduli component of vector bundles is birational to the intermediate Jacobian.
As an applicaton, it is shown that the generic fiber of the period map of Fano
varieties of degree 14 is birational to the intermediate Jacobian of the
associated cubic threefold.Comment: Latex, 28 page
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