algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a

variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''k'' is ruled if it is

birational 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 fu ...

to the product of the projective line with some variety over ''k''. A variety is uniruled if it is covered by a family of

rational 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 c ...

s. (More precisely, a variety ''X'' is uniruled if there is a variety ''Y'' and a dominant rational map ''Y'' × P¹ – → ''X'' which does not factor through the projection to ''Y''.) The concept arose from the

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...

s of 19th-century geometry, meaning surfaces in

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

which are covered by lines. Uniruled varieties can be considered to be relatively simple among all varieties, although there are many of them.

Properties

Every uniruled variety over a field of characteristic zero has

Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...

−∞. The converse is a conjecture which is known in dimension at most 3: a variety of Kodaira dimension −∞ over a field of characteristic zero should be uniruled. A related statement is known in all dimensions: Boucksom, Demailly, Păun and Peternell showed that a

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

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 w ...

''X'' over a field of characteristic zero is uniruled if and only if the

canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...

of ''X'' is not pseudo-effective (that is, not in the closed convex cone spanned by effective divisors in the Néron-Severi group tensored with the real numbers). As a very special case, a smooth

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...

of degree ''d'' in P^''n'' over a field of characteristic zero is uniruled if and only if ''d'' ≤ ''n'', by the

adjunction formula In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded ...

. (In fact, a smooth hypersurface of degree ''d'' ≤ ''n'' in Pⁿ is a

Fano variety In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has ...

and hence is rationally connected, which is stronger than being uniruled.) A variety ''X'' over an

uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

''k'' is uniruled if and only if there is a rational curve passing through every ''k''-point of ''X''. By contrast, there are varieties over the algebraic closure ''k'' of a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

which are not uniruled but have a rational curve through every ''k''-point. (The

Kummer variety In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse. The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface In algebraic geometry, a Kummer quartic su ...

of any non-

supersingular 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 o ...

abelian surface In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bi ...

over _''p'' with ''p'' odd has these properties.) It is not known whether varieties with these properties exist over the algebraic closure of the

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...

. Uniruledness is a

geometric property This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

(it is unchanged under field extensions), whereas ruledness is not. For example, the conic ''x''² + ''y''² + ''z''² = 0 in P² over the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s R is uniruled but not ruled. (The associated curve over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s C is isomorphic to P¹ and hence is ruled.) In the positive direction, every uniruled variety of dimension at most 2 over an algebraically closed field of characteristic zero is ruled. Smooth cubic 3-folds and smooth quartic 3-folds in P⁴ over C are uniruled but not ruled.

Positive characteristic

Uniruledness behaves very differently in positive characteristic. In particular, there are uniruled (and even

unirational 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 :K(U_1, \dots , U_d), t ...

) surfaces of

general type In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...

: an example is the surface ''x''^''p''+1 + ''y''^''p''+1 + ''z''^''p''+1 + ''w''^''p''+1 = 0 in P³ over _''p'', for any prime number ''p'' ≥ 5. So uniruledness does not imply that the Kodaira dimension is −∞ in positive characteristic. A variety ''X'' is separably uniruled if there is a variety ''Y'' with a dominant separable rational map ''Y'' × P¹ – → ''X'' which does not factor through the projection to ''Y''. ("Separable" means that the derivative is surjective at some point; this would be automatic for a dominant rational map in characteristic zero.) A separably uniruled variety has Kodaira dimension −∞. The converse is true in dimension 2, but not in higher dimensions. For example, there is a smooth projective 3-fold over ₂ which has Kodaira dimension −∞ but is not separably uniruled.E. Sato, Tohoku Math. J. 45 (1993), 447-460. Theorem. It is not known whether every smooth Fano variety in positive characteristic is separably uniruled.

Notes

References

* * * * *{{Citation , author1-link=Tetsuji Shioda , last1=Shioda , first1=Tetsuji , title=An example of unirational surfaces in characteristic ''p'' , journal=Mathematische Annalen , volume=211 , year=1974 , pages=233–236 , doi=10.1007/BF01350715 , mr=0374149 Algebraic geometry