Different definitions of ruled surface

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Let $X$ be a projective smooth surface. I read two different definitions of $X$ to be a ruled surface, namely:



  1. $X$ is birational equivalent to some $Btimes mathbb P^1$, where $B$ is projective smooth curve.

  2. Through every point on $X$ there exists a projective line. (I am a little confused about this: this means $X$ is already embedded into some $mathbb P^n$? or means through every point there exists some curve which $cong mathbb P^1$?)

I want to know are these two definitions equivalent?




algebraic-geometry surfaces




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  • Actually this means given any point $pin X$, there exists a non-constant (birational) morphism from $mathbbP^1to X$, whose image contains $p$.
    – Mohan
    Jul 14 at 20:22










  • @Mohan Is there any difference between this and "through every point there exists some curve isomorphic to $mathbb P^1$"? If some curve birational to $mathbb P^1$ then it has to be isomorphic to $mathbb P^1$.
    – Akatsuki
    Jul 14 at 20:59






  • 1




    Singular curves are allowed.
    – Mohan
    Jul 14 at 23:31














up vote
2
down vote

favorite
3












Let $X$ be a projective smooth surface. I read two different definitions of $X$ to be a ruled surface, namely:



  1. $X$ is birational equivalent to some $Btimes mathbb P^1$, where $B$ is projective smooth curve.

  2. Through every point on $X$ there exists a projective line. (I am a little confused about this: this means $X$ is already embedded into some $mathbb P^n$? or means through every point there exists some curve which $cong mathbb P^1$?)

I want to know are these two definitions equivalent?




algebraic-geometry surfaces




share|cite|improve this question





















  • Actually this means given any point $pin X$, there exists a non-constant (birational) morphism from $mathbbP^1to X$, whose image contains $p$.
    – Mohan
    Jul 14 at 20:22










  • @Mohan Is there any difference between this and "through every point there exists some curve isomorphic to $mathbb P^1$"? If some curve birational to $mathbb P^1$ then it has to be isomorphic to $mathbb P^1$.
    – Akatsuki
    Jul 14 at 20:59






  • 1




    Singular curves are allowed.
    – Mohan
    Jul 14 at 23:31












up vote
2
down vote

favorite
3









up vote
2
down vote

favorite
3






3





Let $X$ be a projective smooth surface. I read two different definitions of $X$ to be a ruled surface, namely:



  1. $X$ is birational equivalent to some $Btimes mathbb P^1$, where $B$ is projective smooth curve.

  2. Through every point on $X$ there exists a projective line. (I am a little confused about this: this means $X$ is already embedded into some $mathbb P^n$? or means through every point there exists some curve which $cong mathbb P^1$?)

I want to know are these two definitions equivalent?




algebraic-geometry surfaces




share|cite|improve this question













Let $X$ be a projective smooth surface. I read two different definitions of $X$ to be a ruled surface, namely:



  1. $X$ is birational equivalent to some $Btimes mathbb P^1$, where $B$ is projective smooth curve.

  2. Through every point on $X$ there exists a projective line. (I am a little confused about this: this means $X$ is already embedded into some $mathbb P^n$? or means through every point there exists some curve which $cong mathbb P^1$?)

I want to know are these two definitions equivalent?





algebraic-geometry surfaces





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 14 at 18:37
























asked Jul 14 at 18:30









Akatsuki

8721623




8721623











  • Actually this means given any point $pin X$, there exists a non-constant (birational) morphism from $mathbbP^1to X$, whose image contains $p$.
    – Mohan
    Jul 14 at 20:22










  • @Mohan Is there any difference between this and "through every point there exists some curve isomorphic to $mathbb P^1$"? If some curve birational to $mathbb P^1$ then it has to be isomorphic to $mathbb P^1$.
    – Akatsuki
    Jul 14 at 20:59






  • 1




    Singular curves are allowed.
    – Mohan
    Jul 14 at 23:31
















  • Actually this means given any point $pin X$, there exists a non-constant (birational) morphism from $mathbbP^1to X$, whose image contains $p$.
    – Mohan
    Jul 14 at 20:22










  • @Mohan Is there any difference between this and "through every point there exists some curve isomorphic to $mathbb P^1$"? If some curve birational to $mathbb P^1$ then it has to be isomorphic to $mathbb P^1$.
    – Akatsuki
    Jul 14 at 20:59






  • 1




    Singular curves are allowed.
    – Mohan
    Jul 14 at 23:31















Actually this means given any point $pin X$, there exists a non-constant (birational) morphism from $mathbbP^1to X$, whose image contains $p$.
– Mohan
Jul 14 at 20:22




Actually this means given any point $pin X$, there exists a non-constant (birational) morphism from $mathbbP^1to X$, whose image contains $p$.
– Mohan
Jul 14 at 20:22












@Mohan Is there any difference between this and "through every point there exists some curve isomorphic to $mathbb P^1$"? If some curve birational to $mathbb P^1$ then it has to be isomorphic to $mathbb P^1$.
– Akatsuki
Jul 14 at 20:59




@Mohan Is there any difference between this and "through every point there exists some curve isomorphic to $mathbb P^1$"? If some curve birational to $mathbb P^1$ then it has to be isomorphic to $mathbb P^1$.
– Akatsuki
Jul 14 at 20:59




1




1




Singular curves are allowed.
– Mohan
Jul 14 at 23:31




Singular curves are allowed.
– Mohan
Jul 14 at 23:31















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