Skip to main content
MathOverflow

Return to Answer

While this is on the front page, error in book name; link to answer and comment
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

To complete the answeranswer of DivieriettiDiverietti and the commentcomment of Roy Smith, here is a statement which might interest you:

Theorem If $X,Y$$X$, $Y$ are varieties over a field $k$, assume $X$ is smooth and $Y$ proper containing no rational curves. Then any rational map $X\dashrightarrow Y$ is everywhere defined.

You can find that statement in Debarre's book Higher Dimensional GeometryHigher-Dimensional Algebraic Geometry, Corollary 1.44 p.31 31.

In particular, if $X$ is smooth projective and contains no rational curves, then its automorphism group is equal to the group of its birational endomorphisms.

To complete the answer of Divierietti and the comment of Roy Smith, here is a statement which might interest you:

Theorem If $X,Y$ are varieties over a field $k$, assume $X$ is smooth and $Y$ proper containing no rational curves. Then any rational map $X\dashrightarrow Y$ is everywhere defined.

You can find that statement in Debarre's book Higher Dimensional Geometry, Corollary 1.44 p.31.

In particular, if $X$ is smooth projective and contains no rational curves, then its automorphism group is equal to the group of its birational endomorphisms.

To complete the answer of Diverietti and the comment of Roy Smith, here is a statement which might interest you:

Theorem If $X$, $Y$ are varieties over a field $k$, assume $X$ is smooth and $Y$ proper containing no rational curves. Then any rational map $X\dashrightarrow Y$ is everywhere defined.

You can find that statement in Debarre's book Higher-Dimensional Algebraic Geometry, Corollary 1.44 p. 31.

In particular, if $X$ is smooth projective and contains no rational curves, then its automorphism group is equal to the group of its birational endomorphisms.

Source Link
Henri
Henri
  • 2.7k
  • 1
  • 18
  • 17

To complete the answer of Divierietti and the comment of Roy Smith, here is a statement which might interest you:

Theorem If $X,Y$ are varieties over a field $k$, assume $X$ is smooth and $Y$ proper containing no rational curves. Then any rational map $X\dashrightarrow Y$ is everywhere defined.

You can find that statement in Debarre's book Higher Dimensional Geometry, Corollary 1.44 p.31.

In particular, if $X$ is smooth projective and contains no rational curves, then its automorphism group is equal to the group of its birational endomorphisms.