In mathematics, a supersingular variety is (usually) a smooth

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

in nonzero characteristic such that for all ''n'' the

slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...

s of the

Newton polygon In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was ''essentially'' the field of formal La ...

of the ''n''th

crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by ...

are all ''n''/2 . For special classes of varieties such as elliptic curves it is common to use various ad hoc definitions of "supersingular", which are (usually) equivalent to the one given above. The term "singular elliptic curve" (or "singular ''j''-invariant") was at one times used to refer to complex

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s whose ring of

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...

s has rank 2, the maximum possible.

Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory a ...

discovered that, in finite characteristic, elliptic curves can have larger rings of endomorphisms of rank 4, and these were called "supersingular elliptic curves". Supersingular elliptic curves can also be characterized by the slopes of their crystalline cohomology, and the term "supersingular" was later extended to other varieties whose cohomology has similar properties. The terms "supersingular" or "singular" do not mean that the variety has singularities. Examples include: *

Supersingular elliptic curve In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ''p'' > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular ...

. Elliptic curves in non-zero characteristic with an unusually large ring of endomorphisms of rank 4. *

Supersingular Abelian variety In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all ''n'' the slopes of the Newton polygon of the ''n''th crystalline cohomology are all ''n''/2 . For special classes of ...

Sometimes defined to be an abelian variety isogenous to a product of supersingular elliptic curves, and sometimes defined to be an abelian variety of some rank ''g'' whose endomorphism ring has rank (2''g'')². *

Supersingular K3 surface In algebraic geometry, a supersingular K3 surface is a K3 surface over a field ''k'' of characteristic ''p'' > 0 such that the slopes of Frobenius on the crystalline cohomology ''H''2(''X'',''W''(''k'')) are all equal to 1. These have also been c ...

. Certain K3 surfaces in non-zero characteristic. *

Supersingular Enriques surface In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity ''q'' = 0 and the canonical line bundle ''K'' is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex n ...

. Certain Enriques surfaces in characteristic 2. *A surface is called Shioda supersingular if the rank of its Néron–Severi group is equal to its second Betti number. *A surface is called Artin supersingular if its formal Brauer group has infinite height.

References

* {{Set index article, mathematics Algebraic geometry