728 research outputs found
Rigid affine surfaces with isomorphic A2-cylinders
We construct families of smooth affine surfaces with pairwise non isomorphic
A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as
complements of cuspidal hyperplane sections of smooth projective cubic
surfaces
Affine open subsets in A^3 without the cancellation property
We give families of examples of principal open subsets of the affine space
\mathbb{A}^{3} which do not have the cancellation property. We show as a
by-product that the cylinders over Koras-Russell threefolds of the first kind
have a trivial Makar-Limanov invariant
Complements of hyperplane sub-bundles in projective space bundles over the projective line
We establish that the isomorphy type as an abstract algebraic variety of the
complement of an ample hyperplane sub-bundle H of a projective space bundle of
rank r-1 over the projective line depends only on the the r-fold
self-intersection of H . In particular it depends neither on the ambient bundle
nor on a particular ample hyperplane sub-bundle with given r-fold
self-intersection. Our proof exploits the unexpected property that every such
complement comes equipped with the structure of a non trivial torsor under a
vector bundle on the affine line with a double origin
Log-uniruled affine varieties without cylinder-like open subsets
A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for
smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness,
both properties being in fact equivalent to the negativity of the logarithmic
Kodaira dimension. Here we show in contrast that starting from dimension three,
there exists smooth affine varieties which are affine-uniruled but not
affine-ruled
Real frontiers of fake planes
In [8], we define and partially classify fake real planes, that is, minimal
complex surfaces with conjugation whose real locus is diffeomorphic to the
euclidean real plane . Classification results are given up to
biregular isomorphisms and up to birational diffeomorphisms. In this note, we
describe in an elementary way numerous examples of fake real planes and we
exhibit examples of such planes of every Kodaira dimension which are birationally diffeomorphic to
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