Quartic threefold

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In algebraic geometry, a quartic threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space. Iskovskih & Manin (1971) showed that all non-singular quartic threefolds are irrational, though some of them are unirational.

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Faltingss theorem theorem

In number theory, the Mordell conjecture is the conjecture made by Mordell (1922) that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.

Quartic function Polynomial function of degree four

In algebra, a quartic function is a function of the form

Birational geometry

In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

Cubic surface algebraic surface defined by a single quaternary cubic polynomial which is homogeneous of degree 3

A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial which is homogeneous of degree 3. Cubic surfaces are del Pezzo surfaces.

In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class.

In mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle Ω on V.

In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to

Yuri Manin Russian mathematician

Yuri Ivanovitch Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book "Computable and Uncomputable".

Homological mirror symmetry

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.

In algebraic geometry, a Fano variety, introduced by Gino Fano in, is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities.

In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties Vs. The fibers of the vector bundle are the de Rham cohomology groups of the fibers Vs of the family. It was introduced by Manin (1958) for curves S and by Grothendieck (1966) in higher dimensions.

In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve.

In mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by Burkhardt, with the maximum possible number of 45 nodes.

In algebraic geometry, the Igusa quartic is a quartic hypersurface in 4-dimensional projective space, studied by Igusa (1962). It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the Segre cubic.

In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 dimensional projective space, studied by Corrado Segre (1887).

In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation

In mathematics, a quintic threefold is a degree 5 3-dimensional hypersurface in 4-dimensional projective space. Non-singular quintic threefolds are Calabi–Yau manifolds.

In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation

In mathematics, the André–Oort conjecture is an open problem in number theory that generalises the Manin–Mumford conjecture. A prototypical version of the conjecture was stated by Yves André in 1989 and a more general version was conjectured by Frans Oort in 1995. The modern version is a natural generalisation of these two conjectures.

In mathematics, a Manin triple consists of a Lie algebra g with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p and q such that g is the direct sum of p and q as a vector space.

References

Matematicheskii Sbornik is a peer reviewed Russian mathematical journal founded by the Moscow Mathematical Society in 1866. It is the oldest successful Russian mathematical journal. The English translation is Sbornik: Mathematics. It is also sometimes cited under the alternative name Izdavaemyi Moskovskim Matematicheskim Obshchestvom or its French translation Recueil mathématique de la Société mathématique de Moscou, but the name Recueil mathématique is also used for an unrelated journal, Mathesis. Yet another name, Sovetskii Matematiceskii Sbornik, was listed in a statement in the journal in 1931 apologizing for the former editorship of Dmitri Egorov, who had been recently discredited for his religious views; however, this name was never actually used by the journal.

Digital object identifier Character string used as a permanent identifier for a digital object, in a format controlled by the International DOI Foundation

In computing, a digital object identifier (DOI) is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of Mathematical Reviews and additionally contains citation information for over 3.5 million items as of 2018.