In mathematics, a Raynaud surface is a particular kind of
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
that was introduced in and named for . To be precise, a Raynaud surface is a
quasi-elliptic surface over an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'' greater than 1, such that all fibers are irreducible and the fibration has a section. The
Kodaira vanishing theorem fails for such surfaces; in other words the Kodaira theorem, valid in algebraic geometry over the complex numbers, has such surfaces as counterexamples, and these can only exist in
characteristic ''p''.
Generalized Raynaud surfaces were introduced in , and give examples of surfaces of general type with global vector fields.
References
*
*
*{{Citation , last1=Raynaud , first1=Michel , author1-link=Michel Raynaud , title=C. P. Ramanujam—a tribute , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Tata Inst. Fund. Res. Studies in Math. , mr=541027 , year=1978 , volume=8 , chapter=Contre-exemple au "vanishing theorem" en caractéristique
, pages=273–278
Algebraic surfaces