Difference between revisions of "Bertini theorems"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
Revision as of 21:50, 30 March 2012
Two theorems concerning the properties of linear systems (cf. Linear system) on algebraic varieties, due to E. Bertini [1].
Let be an algebraic variety over an algebraically closed field of characteristic 0, let be a linear system without fixed components on and let be the image of the variety under the mapping given by . The following two theorems are known as the first and the second Bertini theorem, respectively.
1) If , then almost all the divisors of the linear system (i.e. all except a closed subset in the parameter space not equal to ) are irreducible reduced algebraic varieties.
2) Almost all divisors of have no singular points outside the basis points of the linear system and the singular points of the variety .
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 , Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping are irreducible and reduced if the function field is algebraically closed in the field under the imbedding . If the characteristic of is finite, the corresponding theorem is true if the extension 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 |
Bertini theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertini_theorems&oldid=16310