Dolgachev surface
Encyclopedia
In mathematics, Dolgachev surfaces are certain simply connected elliptic surface
s, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic.
X0 of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface Xq is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some q ≥ 3.
The Dolgachev surfaces are simply connected and the bilinear form on the second cohomology group is odd of signature (1, 9) (so it is the unimodular lattice
I1,9). The geometric genus
pg is 0 and the Kodaira dimension
is 1.
found the first examples of homeomorphic but not diffeomorphic 4-manifolds X0 and X3.
More generally the surfaces Xq and Xr are always homeomorphic, but are not diffeomorphic unless q = r.
showed that the Dolgachev surface X3 has a handlebody decomposition without 1- and 3-handles.
Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
s, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic.
Properties
The blowupBlowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point...
X0 of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface Xq is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some q ≥ 3.
The Dolgachev surfaces are simply connected and the bilinear form on the second cohomology group is odd of signature (1, 9) (so it is the unimodular lattice
Unimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
I1,9). The geometric genus
Geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.-Definition:...
pg is 0 and the Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
is 1.
found the first examples of homeomorphic but not diffeomorphic 4-manifolds X0 and X3.
More generally the surfaces Xq and Xr are always homeomorphic, but are not diffeomorphic unless q = r.
showed that the Dolgachev surface X3 has a handlebody decomposition without 1- and 3-handles.