In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the theorem of Bertini is an existence and genericity theorem for smooth connected
hyperplane section In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X'H'' of those elements ''x'' of ''X'' that satisfy the single ...
s for smooth projective varieties over
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 ...
s, introduced by
Eugenio Bertini
Eugenio Bertini (8 November 1846 – 24 February 1933) was an Italian mathematician who introduced Bertini's theorem. He was born at Forlì
Forlì ( , ; rgn, Furlè ; la, Forum Livii) is a ''comune'' (municipality) and city in Emilia-Romagna, ...
. This is the simplest and broadest of the "Bertini theorems" applying to a
linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
These arose first in the fo ...
; simplest because there is no restriction on the
characteristic of the underlying field, while the extensions require characteristic 0.
Statement for hyperplane sections of smooth varieties
Let ''X'' be a smooth quasi-projective variety over an algebraically closed field, embedded in a
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 ...
.
Let
denote the
complete system of hyperplane divisors in
. Recall that it is the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of
and is isomorphic to
.
The theorem of Bertini states that the set of hyperplanes not containing ''X'' and with smooth intersection with ''X'' contains an open dense subset of the total system of divisors
. The set itself is open if ''X'' is projective. If
, then these intersections (called hyperplane sections of ''X'') are connected, hence irreducible.
The theorem hence asserts that a ''general'' hyperplane section not equal to ''X'' is smooth, that is: the property of smoothness is generic.
Over an arbitrary field ''k'', there is a dense open subset of the dual space
whose
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
s define hyperplanes smooth hyperplane sections of ''X''. When ''k'' is infinite, this open subset then has infinitely many rational points and there are infinitely many smooth hyperplane sections in ''X''.
Over a finite field, the above open subset may not contain rational points and in general there is no hyperplanes with smooth intersection with ''X''. However, if we take hypersurfaces of sufficiently big degrees, then the theorem of Bertini holds.
Outline of a proof
We consider the subfibration of the product variety
with fiber above
the linear system of hyperplanes that intersect ''X'' non-
transversally at ''x''.
The rank of the fibration in the product is one less than the codimension of
, so that the total space has lesser dimension than
and so its projection is contained in a divisor of the complete system
.
General statement
Over any infinite field
of characteristic 0, if ''X'' is a smooth quasi-projective
-variety, a general member of a
linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
These arose first in the fo ...
on ''X'' is smooth away from the
base locus
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
These arose first in the fo ...
of the system. For clarification, this means that given a linear system
, the preimage
of a hyperplane ''H'' is smooth -- outside the base locus of ''f'' -- for all hyperplanes ''H'' in some dense open subset of the dual projective space
. This theorem also holds in characteristic p>0 when the linear system ''f'' is unramified.
Generalizations
The theorem of Bertini has been generalized in various ways. For example, a result due to
Steven Kleiman
Steven Lawrence Kleiman (born March 31, 1942) is an American mathematician.
Professional career
Kleiman is a Professor of Mathematics at the Massachusetts Institute of Technology. Born in Boston, he did his undergraduate studies at MIT. He rece ...
asserts the following (cf.
Kleiman's theorem In algebraic geometry, Kleiman's theorem, introduced by , concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection.
Precisely, it states: given a connected algebraic group ''G'' acti ...
): for a connected
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
''G'', and any
homogeneous ''G''-variety ''X'', and two varieties ''Y'' and ''Z'' mapping to ''X'', let ''Y''
σ be the variety obtained by letting σ ∈ ''G'' act on ''Y''. Then, there is an open dense subscheme ''H'' of ''G'' such that for σ ∈ ''H'',
is either empty or purely of the (expected)
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
dim ''Y'' + dim ''Z'' − dim ''X''. If, in addition, ''Y'' and ''Z'' are
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 algebrai ...
and the base field has characteristic zero, then ''H'' may be taken such that
is smooth for all
, as well. The above theorem of Bertini is the special case where
is expressed as the quotient of SL
''n'' by the
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
of upper triangular matrices, ''Z'' is a subvariety and ''Y'' is a hyperplane.
Theorem of Bertini has also been generalized to discrete valuation domains or finite fields, or for étale coverings of ''X''.
The theorem is often used for induction steps.
See also
*
Grothendieck's connectedness theorem In mathematics, Grothendieck's connectedness theorem , states that if ''A'' is a complete Noetherian local ring whose spectrum is ''k''-connected and ''f'' is in the maximal ideal, then Spec(''A''/''fA'') is (''k'' − 1)-connected. ...
Notes
References
* {{Hartshorne AG
Bertini and his two fundamental theoremsby Steven L. Kleiman, on the life and works of
Eugenio Bertini
Eugenio Bertini (8 November 1846 – 24 February 1933) was an Italian mathematician who introduced Bertini's theorem. He was born at Forlì
Forlì ( , ; rgn, Furlè ; la, Forum Livii) is a ''comune'' (municipality) and city in Emilia-Romagna, ...
Geometry of divisors
Theorems in algebraic geometry