Suppose $X$ is a projective variety and $D$ is a smooth divisor and let $L = \mathcal{O}(D)$ be the line bundle corresponding to $D$. Consider $X \times \mathbb{P}^1$ with the line bundle $\mathcal{O}(D) \otimes \mathcal{O}(1)$. Assume that this line bundle has a section such that the zero set is a smooth divisor $D'$. Is $D'$ well-studied in algebraic geometry, topology, ...?
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2$\begingroup$ Probably I do not understand your question. Clearly $D'$ is a section of the trivial bundle $\pi \colon X \times {\mathbb P}^1 \rightarrow X$. So $D'$ is isomorphic to $X$. On the other hand, every section of the bundle is a divisor in $\pi^*L \otimes {\mathcal O}(1)$ for some line bundle $L$ on $X$. $\endgroup$– Roberto PignatelliJul 10, 2014 at 16:45
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$\begingroup$ Hi Roberto, it is not $X$. For example let $s$ be section of $L$ such that $s^{-1}(0) = D$ and $z_0$ be a section of $\mathcal{O}(1)$ on $\mathbb{P}^1$. Then the zero section of $z_0s$ is $X \times [0:1] \cup D \times \mathbb{P}^1$. So $D'$ should be a smoothing out of this guy. $\endgroup$– klnJul 10, 2014 at 18:37
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$\begingroup$ OK, you are right, I take it back, I was thinking to the case when $X$ is a curve. Indeed, the intersection of $D'$ with a fibre of $\pi$ is either a reduced point or the whole fibre. If $\dim X=1$, the latter case can't occur since you assume $X$ smooth, but in bigger dimension I can only conclude that $D'$ is mapped birationally to $X$ contracting some ${\mathbb P}^1$. $\endgroup$– Roberto PignatelliJul 10, 2014 at 18:50
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$\begingroup$ If $\dim X=2$ we are blowing up some points, the classical construction of the blow up of ${\mathbb P}^2$ in a point is in this way. In the higher dimensional case more complicated contractions can occur. Sorry, I should have thought more on your question before posting. $\endgroup$– Roberto PignatelliJul 10, 2014 at 18:51
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$\begingroup$ A section of $O(D)\otimes O(1)$ can be thought of as a pencil of sections of $O(D)$. Then $D'$ is "the total space" of this pencil. If smooth, it is isomorphic to the blowup of $X$ in the base locus of the pencil. $\endgroup$– SashaJul 13, 2014 at 15:50
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