Timeline for On the intersection numbers of the generators of $\text{Pic}(X)$ of a smooth quintic surface
Current License: CC BY-SA 4.0
14 events
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| Jul 2, 2021 at 17:04 | history | edited | Yusuf Mustopa | CC BY-SA 4.0 |
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| Jul 2, 2021 at 12:40 | comment | added | Yusuf Mustopa | You're welcome! This general result is contained in the monograph of Lopez cited in the answer. | |
| Jul 2, 2021 at 6:30 | comment | added | User | thank you very much for the answer. Can you give me a reference of the general result you just mentioned? | |
| Jul 2, 2021 at 6:03 | comment | added | Yusuf Mustopa | See the edit above. | |
| Jul 2, 2021 at 6:03 | history | edited | Yusuf Mustopa | CC BY-SA 4.0 |
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| Jun 28, 2021 at 7:00 | comment | added | User | In this context I have one further question : Does it make sense to expect the existence of a smooth quintic hypersurface $X$ in $\mathbb P^3$ containing a line $L$ (even if this is rare) such that $\text{Pic}(X)$ is generated by the hyperplane section $H$ and the line $L$? | |
| May 26, 2020 at 17:38 | vote | accept | User | ||
| May 26, 2020 at 17:14 | comment | added | Yusuf Mustopa | As to your second comment, you are correct that the smooth surfaces in under discussion are not general when $d \geq 4.$ | |
| May 26, 2020 at 17:12 | comment | added | Yusuf Mustopa | The curve $C$ is cut out by the maximal minors of a $d \times (d-1)$-submatrix of the matrix of linear forms cutting out $X,$ so by general results on determinantal varieties we can ensure $C$ is smooth and connected, and therefore irreducible. The degree $C \cdot H$ and genus can of course be read off the Hilbert polynomial, which in turn can be read off the Eagon-Northcott resolution of the associated determinantal ideal. You can then determine $C \cdot C$ from the adjunction formula on $X.$ I can't access the reference at the moment, so I can't speak to the relevance of Theorem (II)3. | |
| May 26, 2020 at 14:26 | comment | added | User | also as the general surface of degree $d \geq4$ is not linear determinantal, but linear pfaffian if $d \leq 15$, so in this case are we talking about smooth quintic surfaces which is not general? | |
| May 26, 2020 at 11:49 | comment | added | User | thanks for the answer. Is the curve $C$ mentioned in the answer irreducible? Are the intersection numbers $C.C$ and $C.H$ immediate from the discussion? (apology in advance, if this is trivial to compute). Finally, Is theorem $(II)3.1$ mentioned there relevant in this context? | |
| May 25, 2020 at 18:52 | history | edited | Yusuf Mustopa | CC BY-SA 4.0 |
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| May 25, 2020 at 18:26 | history | edited | Yusuf Mustopa | CC BY-SA 4.0 |
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| May 25, 2020 at 17:19 | history | answered | Yusuf Mustopa | CC BY-SA 4.0 |