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Two theorems concerning the properties of linear systems (cf. [[Linear system|Linear system]]) on algebraic varieties, due to E. Bertini [[#References|[1]]].
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Two theorems concerning the properties of [[linear system]]s on algebraic varieties, due to E. Bertini [[#References|[1]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157701.png" /> be an algebraic variety over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157702.png" /> of characteristic 0, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157703.png" /> be a linear system without fixed components on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157704.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157705.png" /> be the image of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157706.png" /> under the mapping given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157707.png" />. The following two theorems are known as the first and the second Bertini theorem, respectively.
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Let $V$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0, let $L$ be a linear system without fixed components on $V$ and let $W$ be the image of the variety $V$ under the mapping given by $L$. The following two theorems are known as the first and the second Bertini theorem, respectively.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157708.png" />, then almost all the divisors of the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b0157709.png" /> (i.e. all except a closed subset in the parameter space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577010.png" /> not equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577011.png" />) are irreducible reduced algebraic varieties.
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1) If $\dim W > 1$, then almost all the divisors of the linear system $L$ (i.e. all except a closed subset in the parameter space $P(L)$ not equal to $P(L)$) are irreducible reduced algebraic varieties.
  
2) Almost all divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577012.png" /> have no singular points outside the basis points of the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577013.png" /> and the singular points of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577014.png" />.
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2) Almost all divisors of $L$ have no singular points outside the basis points of the linear system $L$ and the singular points of the variety $V$>.
  
 
Both Bertini theorems are invalid if the characteristic of the field is non-zero.
 
Both Bertini theorems are invalid if the characteristic of the field is non-zero.
  
Conditions under which Bertini's theorems are valid for the case of a finite characteristic of the field have been studied [[#References|[3]]], [[#References|[6]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577015.png" />, Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577016.png" /> are irreducible and reduced if the function field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577017.png" /> is algebraically closed in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577018.png" /> under the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577019.png" />. If the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577020.png" /> is finite, the corresponding theorem is true if the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015770/b01577021.png" /> is separable [[#References|[3]]], [[#References|[6]]]. The Bertini theorems apply to linear systems of hyperplane sections, without restrictions on the characteristic of the field [[#References|[5]]].
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Conditions under which Bertini's theorems are valid for the case of a finite characteristic of the field have been studied [[#References|[3]]], [[#References|[6]]]. If $\dim W = 1$, Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping $\phi-L : V \to W$ are irreducible and reduced if the function field $k(W)$ is algebraically closed in the field $k(V)$ under the imbedding $\phi_L^* : k(W) \to k(V)$. If the characteristic of $k$ is finite, the corresponding theorem is true if the extension $k(V)/k(W)$ is [[Separable extension|separable]] [[#References|[3]]], [[#References|[6]]]. The Bertini theorems apply to linear systems of hyperplane sections, without restrictions on the characteristic of the field [[#References|[5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Bertini, "Introduction to the projective geometry of hyperspaces" , Messina (1923) (In Italian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian) {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> Y. Akizuki, "Theorems of Bertini on linear systems" ''J. Math. Soc. Japan'' , '''3''' : 1 (1951) pp. 170–180 {{MR|0044160}} {{ZBL|0043.36302}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Y. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" ''Mem. Coll. Sci. Univ. Kyoto Ser. A Math.'' , '''26''' : 2 (1950) pp. 185–187 {{MR|0044161}} {{ZBL|0045.42001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> O. Zariski, "The theorem of Bertini on the variable singular points of a linear system of varieties" ''Trans. Amer. Math. Soc.'' , '''56''' (1944) pp. 130–140 {{MR|0011572}} {{ZBL|0061.33101}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> E. Bertini, "Introduction to the projective geometry of hyperspaces" , Messina (1923) (In Italian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian) {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top"> Y. Akizuki, "Theorems of Bertini on linear systems" ''J. Math. Soc. Japan'' , '''3''' : 1 (1951) pp. 170–180 {{MR|0044160}} {{ZBL|0043.36302}} </TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top"> Y. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" ''Mem. Coll. Sci. Univ. Kyoto Ser. A Math.'' , '''26''' : 2 (1950) pp. 185–187 {{MR|0044161}} {{ZBL|0045.42001}} </TD></TR>
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<TR><TD valign="top">[6]</TD> <TD valign="top"> O. Zariski, "The theorem of Bertini on the variable singular points of a linear system of varieties" ''Trans. Amer. Math. Soc.'' , '''56''' (1944) pp. 130–140 {{MR|0011572}} {{ZBL|0061.33101}} </TD></TR>
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<TR><TD valign="top">[7]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 11:57, 23 April 2017

Two theorems concerning the properties of linear systems on algebraic varieties, due to E. Bertini [1].

Let $V$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0, let $L$ be a linear system without fixed components on $V$ and let $W$ be the image of the variety $V$ under the mapping given by $L$. The following two theorems are known as the first and the second Bertini theorem, respectively.

1) If $\dim W > 1$, then almost all the divisors of the linear system $L$ (i.e. all except a closed subset in the parameter space $P(L)$ not equal to $P(L)$) are irreducible reduced algebraic varieties.

2) Almost all divisors of $L$ have no singular points outside the basis points of the linear system $L$ and the singular points of the variety $V$>.

Both Bertini theorems are invalid if the characteristic of the field is non-zero.

Conditions under which Bertini's theorems are valid for the case of a finite characteristic of the field have been studied [3], [6]. If $\dim W = 1$, Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping $\phi-L : V \to W$ are irreducible and reduced if the function field $k(W)$ is algebraically closed in the field $k(V)$ under the imbedding $\phi_L^* : k(W) \to k(V)$. If the characteristic of $k$ is finite, the corresponding theorem is true if the extension $k(V)/k(W)$ is separable [3], [6]. The Bertini theorems apply to linear systems of hyperplane sections, without restrictions on the characteristic of the field [5].

References

[1] E. Bertini, "Introduction to the projective geometry of hyperspaces" , Messina (1923) (In Italian)
[2] "Algebraic surfaces" Trudy Mat. Inst. Steklov. , 75 (1965) (In Russian) Zbl 0154.33002 Zbl 0154.21001
[3] M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902
[4] Y. Akizuki, "Theorems of Bertini on linear systems" J. Math. Soc. Japan , 3 : 1 (1951) pp. 170–180 MR0044160 Zbl 0043.36302
[5] Y. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" Mem. Coll. Sci. Univ. Kyoto Ser. A Math. , 26 : 2 (1950) pp. 185–187 MR0044161 Zbl 0045.42001
[6] O. Zariski, "The theorem of Bertini on the variable singular points of a linear system of varieties" Trans. Amer. Math. Soc. , 56 (1944) pp. 130–140 MR0011572 Zbl 0061.33101
[7] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Bertini theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertini_theorems&oldid=23762
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article