Timeline for Curves on varieties, and a criterion for nef divisor
Current License: CC BY-SA 3.0
9 events
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| Nov 15, 2014 at 14:45 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jul 30, 2013 at 11:46 | comment | added | Francesco Polizzi | If one has two irreducible curves $C$, $D$ on a smooth complex projective surface, with $C \neq D$, then $C \cdot D$ is the number of intersections of $C$ and $D$ (counted with multiplicities). In particular $C \cdot D \geq 0$. Then if $C \cdot D <0$ it follows that $C$ and $D$ necessarily have some component in common (and such component has negative self-intersection). This explains (1) and also (2). Notice that in this example one does not need to know all the effective divisors on $X$. | |
| Jul 30, 2013 at 10:50 | comment | added | Li Yutong | Thank you so much! I have two further questions regard your answer: (1) Why you claim " since $C\cdot D=-1$, ..., it follows that $C$ is a component of D" ?(2) Why $C$ has nonnegative intersection with any irreducible curve differ from $C$? I think the two questions are essentially the same: how to describe the effective divisors on $X$? | |
| Jul 30, 2013 at 10:38 | vote | accept | Li Yutong | ||
| Jul 30, 2013 at 10:31 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jul 30, 2013 at 10:10 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jul 30, 2013 at 10:03 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jul 30, 2013 at 9:44 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jul 30, 2013 at 9:37 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |