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A question on the base change of elliptic threefolds
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Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $sigma_i$ ,$i=1 dots n$. None of these (in this specific example) "sections" are honestly a section, they are actually birational to the base manifold rather than being isomorphic. More precisely, they wrap around a finite number of (-1)-curves, so they correspond to a blow of the base at some smooth points.
Let's blow up the base at those finite number of points, such that at least one of the sections, say $sigma_1$, becomes isomorphic to the base. So basically, we define a new elliptically fibered threefold via base change $tildeX = X times_B sigma_1$.
What I want to do is to compute the intersection of divisors after the blow up in terms of the information I have in X.
For example, suppose $sigma_1$ wrap around only one (-1)-curve, I know $sigma_1$ satisfy a relation like,
$sigma_1^2=-c_1(B)cdot sigma_1 + e$,
where $c_1(B)$ is the first Chern class of the base, and $e$ is a codimension 2 cycle, which actually correspond to the (-1)-curve that $sigma_1$ wraps around. (roughly speaking we can find a relation between $e$ and $sigma_i cdot D_b$ and $D_b^2$ ($D_b$ is a base divisor))
So, if $tildesigma_1$ is the corresponding section in $tildeX$, what can be said about,
$tildesigma_1 cdot tildesigma_1=?$.
I did some calculations which seem pretty reasonable to me, but the final answers are not satisfying. For example, I compute the Euler number of $tildesigma_1$, and it's not correct.
I cannot get into more details, because it will become too lengthy. I would appreciate if someone can give me a hint, for example how should I compute $tildesigma_1 cdot tildesigma_1$.
algebraic-geometry birational-geometry
asked Aug 6 at 15:13
Mohsen Karkheiran
466
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Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $sigma_i$ ,$i=1 dots n$. None of these (in this specific example) "sections" are honestly a section, they are actually birational to the base manifold rather than being isomorphic. More precisely, they wrap around a finite number of (-1)-curves, so they correspond to a blow of the base at some smooth points.
Let's blow up the base at those finite number of points, such that at least one of the sections, say $sigma_1$, becomes isomorphic to the base. So basically, we define a new elliptically fibered threefold via base change $tildeX = X times_B sigma_1$.
What I want to do is to compute the intersection of divisors after the blow up in terms of the information I have in X.
For example, suppose $sigma_1$ wrap around only one (-1)-curve, I know $sigma_1$ satisfy a relation like,
$sigma_1^2=-c_1(B)cdot sigma_1 + e$,
where $c_1(B)$ is the first Chern class of the base, and $e$ is a codimension 2 cycle, which actually correspond to the (-1)-curve that $sigma_1$ wraps around. (roughly speaking we can find a relation between $e$ and $sigma_i cdot D_b$ and $D_b^2$ ($D_b$ is a base divisor))
So, if $tildesigma_1$ is the corresponding section in $tildeX$, what can be said about,
$tildesigma_1 cdot tildesigma_1=?$.
I did some calculations which seem pretty reasonable to me, but the final answers are not satisfying. For example, I compute the Euler number of $tildesigma_1$, and it's not correct.
I cannot get into more details, because it will become too lengthy. I would appreciate if someone can give me a hint, for example how should I compute $tildesigma_1 cdot tildesigma_1$.
algebraic-geometry birational-geometry
asked Aug 6 at 15:13
Mohsen Karkheiran
466
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $sigma_i$ ,$i=1 dots n$. None of these (in this specific example) "sections" are honestly a section, they are actually birational to the base manifold rather than being isomorphic. More precisely, they wrap around a finite number of (-1)-curves, so they correspond to a blow of the base at some smooth points.
Let's blow up the base at those finite number of points, such that at least one of the sections, say $sigma_1$, becomes isomorphic to the base. So basically, we define a new elliptically fibered threefold via base change $tildeX = X times_B sigma_1$.
What I want to do is to compute the intersection of divisors after the blow up in terms of the information I have in X.
For example, suppose $sigma_1$ wrap around only one (-1)-curve, I know $sigma_1$ satisfy a relation like,
$sigma_1^2=-c_1(B)cdot sigma_1 + e$,
where $c_1(B)$ is the first Chern class of the base, and $e$ is a codimension 2 cycle, which actually correspond to the (-1)-curve that $sigma_1$ wraps around. (roughly speaking we can find a relation between $e$ and $sigma_i cdot D_b$ and $D_b^2$ ($D_b$ is a base divisor))
So, if $tildesigma_1$ is the corresponding section in $tildeX$, what can be said about,
$tildesigma_1 cdot tildesigma_1=?$.
I did some calculations which seem pretty reasonable to me, but the final answers are not satisfying. For example, I compute the Euler number of $tildesigma_1$, and it's not correct.
I cannot get into more details, because it will become too lengthy. I would appreciate if someone can give me a hint, for example how should I compute $tildesigma_1 cdot tildesigma_1$.
algebraic-geometry birational-geometry
asked Aug 6 at 15:13
Mohsen Karkheiran
466
Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $sigma_i$ ,$i=1 dots n$. None of these (in this specific example) "sections" are honestly a section, they are actually birational to the base manifold rather than being isomorphic. More precisely, they wrap around a finite number of (-1)-curves, so they correspond to a blow of the base at some smooth points.
Let's blow up the base at those finite number of points, such that at least one of the sections, say $sigma_1$, becomes isomorphic to the base. So basically, we define a new elliptically fibered threefold via base change $tildeX = X times_B sigma_1$.
What I want to do is to compute the intersection of divisors after the blow up in terms of the information I have in X.
For example, suppose $sigma_1$ wrap around only one (-1)-curve, I know $sigma_1$ satisfy a relation like,
$sigma_1^2=-c_1(B)cdot sigma_1 + e$,
where $c_1(B)$ is the first Chern class of the base, and $e$ is a codimension 2 cycle, which actually correspond to the (-1)-curve that $sigma_1$ wraps around. (roughly speaking we can find a relation between $e$ and $sigma_i cdot D_b$ and $D_b^2$ ($D_b$ is a base divisor))
So, if $tildesigma_1$ is the corresponding section in $tildeX$, what can be said about,
$tildesigma_1 cdot tildesigma_1=?$.
I did some calculations which seem pretty reasonable to me, but the final answers are not satisfying. For example, I compute the Euler number of $tildesigma_1$, and it's not correct.
I cannot get into more details, because it will become too lengthy. I would appreciate if someone can give me a hint, for example how should I compute $tildesigma_1 cdot tildesigma_1$.
algebraic-geometry birational-geometry
asked Aug 6 at 15:13
Mohsen Karkheiran
466
edited Aug 6 at 15:41
asked Aug 6 at 15:13
Mohsen Karkheiran
466
asked Aug 6 at 15:13
Mohsen Karkheiran
466
asked Aug 6 at 15:13
Mohsen Karkheiran
466
466
add a comment |Â
add a comment |Â
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