Timeline for Question on Kähler/ample cone, cone of curves....
Current License: CC BY-SA 2.5
12 events
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| Feb 21, 2011 at 3:44 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Feb 16, 2011 at 21:50 | comment | added | Mohammad Farajzadeh-Tehrani | Two more questions: 1-Consider the example of C.Y 3-fold $X$ in $\mathbb{P}^1 \times \mathbb{P}^3$ which is a double cover over $\P^3$ and $K3$ fibration over $\P^1$. In this case, can you tell me what is $NE(X)$ or $\bar{NE}(X)$ ? I can see two curves in it. One the generator of cone of a $K3$ fiber and the other one, the pull-back of a line from $K3$. ($X$ has the properties you mentioned in answering Q2) 2- In simply connected situation as before, is Kahler cone the dual of $NE(X)$ .i.e. is $\bar\mathfrak{K}_X=$ set of $ w\in H^2(X)$ such that $w(C)\geq 0$ for any $C\in \bar{NE}(X)$? | |
| Feb 16, 2011 at 17:24 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Feb 16, 2011 at 16:50 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Feb 16, 2011 at 16:42 | comment | added | Sándor Kovács | The Picard number (the rank of the Picard group as an abelian group) of a general K3 or more generally CY manifold is 1. The rank of $NE(X)$ is equal the Picard number. For explicit examples consider hypersurfaces a general hypersurface $X\subset \mathbb P^n$ of degree $n+1$ is a CY manifold and by the Lefschetz hyperplane theorem (for n=3 Noether's theorem) its Picard group is isomorphic to the Picard group of the ambient space, which is $\mathbb Z$. | |
| Feb 16, 2011 at 16:40 | comment | added | Sándor Kovács | Artie, thanks for pointing that out, I mixed up things about curves and divisors. I'll try to clean this up.... | |
| Feb 16, 2011 at 8:11 | comment | added | user5117 | Dear S\'andor, about your answer to 1): what is "that map"? | |
| Feb 16, 2011 at 1:56 | comment | added | Mohammad Farajzadeh-Tehrani | About Q2: Can you say why $NE(X)$ is one dimensional for a generic projective $K3$. What if $X$ is a C.Y 3-fold? | |
| Feb 16, 2011 at 1:54 | vote | accept | Mohammad Farajzadeh-Tehrani | ||
| Feb 16, 2011 at 0:48 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Feb 16, 2011 at 0:42 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Feb 16, 2011 at 0:37 | history | answered | Sándor Kovács | CC BY-SA 2.5 |