Subthreshold slope - Revision history

en>Rpeterson1991 at 18:18, 1 November 2014

2014-11-01T18:18:14Z

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	~~In [[mathematics]], the '''theorem of Bertini'''~~ is ~~an existence~~ and ~~genericity theorem for smooth connected [[hyperplane section]]s for smooth projective varieties over [[algebraically closed field]]s, introduced by [[Eugenio Bertini]]. This is the simplest~~ and broadest of the "Bertini theorems" applying to a [[linear system of divisors]]; simplest because there is no restriction on the [[characteristic of a field\|characteristic]] of the underlying field, while the extensions require characteristic 0.<ref>{{Springer\|id=B/b015770\|title=Bertini theorems}}</ref><ref>Hartshorne, Ch. III.10.</ref>		My name is Tayla and I am studying Physical and Agriculture and Life Sciences at Sinagra / Australia.<br><br>Feel free to surf to my website ... [https://www.facebook.com/pages/Kim-Kardashian-Hollywood-Hack-Cheats-Tool/723317271055251 kim kardashian hollywood hack tool]

	~~== Statement for hyperplane sections of smooth varieties==~~
	~~Let ''X'' be a smooth quasi-projective variety over an algebraically closed field, embedded in a [[algebraic geometry of projective spaces\|projective space]] <math>\mathbf P^n<~~/~~math>~~.
	~~Let <math>\|H\|</math> denote the [[linear system of divisors\|complete system]] of hyperplane divisors in~~ <~~math~~>~~\mathbf P^n~~<~~/math>. Recall that it is the [[Duality_(projective_geometry)#Higher dimensional duality\|dual space]] <math>(\mathbf P^n)^{\star}</math> of <math>\mathbf P^n</math~~> ~~and is isomorphic~~ to ~~<math>\mathbf P^n</math>.~~

	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 <math>\|H\|</math>. The set itself is open if ''X'' is projective. If dim(''X'') ≥ 2, 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 <math>(\mathbf P^n)^{\star}</math> whose [[rational point]]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 sufficientely big degrees, then the theorem of Bertini holds~~.~~<ref>Bjorn Poonen:~~ [~~http~~://~~math~~.~~mit~~.~~edu~~/~~~poonen/papers~~/~~bertini.pdf ''Bertini Theorems over finite fields''], Ann. of Math. '''160''' (2004).</ref>~~

	~~==Outline of a proof==~~
	~~We consider the subfibration of the product variety <math>X \times \|H\|</math> with fiber above <math>x\in X</math> the linear system of hyperplanes that intersect ''X'' non~~-~~[[transversality (mathematics)\|transversally]] at ''x''.~~

	The rank of the fibration in the product is one less than the codimension of <math>X \subset \mathbf P^n</math>, so that the total space has lesser dimension than <math>n</math> and so its projection is contained in a divisor of the complete system <math>\|H\|</math>.

	~~==General statement==~~
	~~Over <math>\mathbb C</math> (or an algebraically closed field of characteristic 0), if ''X'' is a smooth quasi~~-~~projective variety, a general member of a [[linear system of divisors]] on ''X'' is smooth away from the [[base locus]] of the system. This property fails in positive characteristics.~~

	~~== Generalizations ==~~
	~~The theorem of Bertini has been generalized in various ways. For example, a result due to [[Steven Kleiman]] asserts the following: for a connected [[algebraic group]] ''G'', and any homogeneous ''G''~~-variety ''X'', and two varieties ''Y'' and ''Z'' mapping to ''X'', let ''Y''<sup>σ</sup> 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'', <math>Y^\sigma \times_X Z</math> is either empty or purely of the (expected) [[Krull dimension\|dimension]] dim ''Y'' + dim ''Z'' − dim ''X''. If, in addition, ''Y'' and ''Z'' are [[smooth scheme\|smooth]], then ''H'' may be taken such that <math>Y^\sigma \times_X Z</math> is smooth for all <math>\sigma \in H</math>, as well. The above theorem of Bertini is the special case where <math>X = \mathbb P^n</math> is expressed as the quotient of SL<sub>''n''</sub> by the [[parabolic subgroup]] of upper triangular matrices, ''Z'' is a subvariety and ''Y'' is a hyperplane.<ref>{{Citation \| last1=Kleiman \| first1=Steven L. \| author1-link=Steven Kleiman \| title=The transversality of a general translate \| year=1974 \| journal=Compositio Mathematica \| issn=0010-~~437X \| volume=28 \| pages=287–297}}</ref>~~

	~~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.~~

	~~== Notes ==~~
	~~{{Reflist}}~~

	~~== References ==~~
	* {{Hartshorne AG}}
	*[http://arxiv.org/pdf/alg-~~geom~~/~~9704018v1 Bertini and his two fundamental theorems] by Steven L. Kleiman, on the life and works of [[Eugenio Bertini]]~~

	~~[[Category:Geometry of divisors]]~~
	~~[[Category:Theorems in algebraic geometry]~~]

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2014-01-13T03:39:19Z

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