View source for Bertini theorems

Two theorems concerning the properties of [[linear system]]s on algebraic varieties, due to E. Bertini [[#References|[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 [[#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====
<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>

{{TEX|done}}