Timeline for Negative curves on surface of general type
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 24, 2017 at 6:36 | comment | added | abx | $\pi$ induces on each component a one-to-one map onto the (smooth) curve downstairs. That map is necessarily an isomorphism (think of the normalization). | |
| Dec 24, 2017 at 6:29 | comment | added | Feng Hao | Sorry I have one more issue. Why have the two components to be smooth? , since it is branched covering. | |
| Dec 24, 2017 at 6:15 | history | edited | abx | CC BY-SA 3.0 |
added 218 characters in body
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| Dec 24, 2017 at 6:13 | comment | added | Feng Hao | Sorry let me add my comments again: Thank you for your example Prof. Beauville. For the negative curve I always assume it is integral. I am not worrying about the geometric genus going to infinity, since I can prove that for surface of general type with ample canonical line bundle, there are only finitely many negative integral curves with geometric genus less than any given integer. I was worrying about that $C_i^2$ can be a fixed number. Are almost all the pullback curve $\pi^*E_n$ irreducibe in $X$? Thank you. | |
| Dec 24, 2017 at 6:11 | comment | added | abx | It is not clear indeed, but I think it follows from the fact that $X$ contains only finitely many smooth rational curves (Lu-Miyaoka). If $\pi ^*E_n$ is not irreducible, it is the union of two smooth rational curves. I will edit my answer. | |
| Dec 24, 2017 at 6:11 | vote | accept | Feng Hao | ||
| Dec 24, 2017 at 5:24 | history | answered | abx | CC BY-SA 3.0 |