Local property of a flat family of schemes

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Let $pi: X to S$ be a flat, smooth family of genus $0$ curves, for example. Now take a point $x in S$, then $x in pi{-1}(p)$ from some $p in S$, i.e $x in mathbb{P}^1$.



I am trying to relate such families to first order infinitesimal deformations.



In particular, does there exist an open subset $ xin U subset X$ such that $U$ is a first order infinitesimal deformation of $mathbb{P}^1$?










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  • $begingroup$
    In your first paragraph, $x in S$ and (presumably) $x in pi^{-1}(p)$, which doesn't make sense.
    $endgroup$
    – RghtHndSd
    Dec 12 '18 at 3:31


















1












$begingroup$


Let $pi: X to S$ be a flat, smooth family of genus $0$ curves, for example. Now take a point $x in S$, then $x in pi{-1}(p)$ from some $p in S$, i.e $x in mathbb{P}^1$.



I am trying to relate such families to first order infinitesimal deformations.



In particular, does there exist an open subset $ xin U subset X$ such that $U$ is a first order infinitesimal deformation of $mathbb{P}^1$?










share|cite|improve this question









$endgroup$












  • $begingroup$
    In your first paragraph, $x in S$ and (presumably) $x in pi^{-1}(p)$, which doesn't make sense.
    $endgroup$
    – RghtHndSd
    Dec 12 '18 at 3:31
















1












1








1





$begingroup$


Let $pi: X to S$ be a flat, smooth family of genus $0$ curves, for example. Now take a point $x in S$, then $x in pi{-1}(p)$ from some $p in S$, i.e $x in mathbb{P}^1$.



I am trying to relate such families to first order infinitesimal deformations.



In particular, does there exist an open subset $ xin U subset X$ such that $U$ is a first order infinitesimal deformation of $mathbb{P}^1$?










share|cite|improve this question









$endgroup$




Let $pi: X to S$ be a flat, smooth family of genus $0$ curves, for example. Now take a point $x in S$, then $x in pi{-1}(p)$ from some $p in S$, i.e $x in mathbb{P}^1$.



I am trying to relate such families to first order infinitesimal deformations.



In particular, does there exist an open subset $ xin U subset X$ such that $U$ is a first order infinitesimal deformation of $mathbb{P}^1$?







algebraic-geometry commutative-algebra deformation-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Dec 4 '18 at 20:01









JadwigaJadwiga

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  • $begingroup$
    In your first paragraph, $x in S$ and (presumably) $x in pi^{-1}(p)$, which doesn't make sense.
    $endgroup$
    – RghtHndSd
    Dec 12 '18 at 3:31




















  • $begingroup$
    In your first paragraph, $x in S$ and (presumably) $x in pi^{-1}(p)$, which doesn't make sense.
    $endgroup$
    – RghtHndSd
    Dec 12 '18 at 3:31


















$begingroup$
In your first paragraph, $x in S$ and (presumably) $x in pi^{-1}(p)$, which doesn't make sense.
$endgroup$
– RghtHndSd
Dec 12 '18 at 3:31






$begingroup$
In your first paragraph, $x in S$ and (presumably) $x in pi^{-1}(p)$, which doesn't make sense.
$endgroup$
– RghtHndSd
Dec 12 '18 at 3:31












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