Timeline for What is the maximal Picard number of a surface in P^3?
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11 events
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| Jun 29, 2014 at 3:00 | comment | added | Noam D. Elkies | @Jason Starr Smooth quartics are K3, and (in characteristic zero) their Picard number attains its maximum of $h^{1,1} = 20$. Segre's quartic has more lines than the Fermat quartic but both have maximal Picard number. [Segre's surface is $T(X,Y) = T(X',Y')$ where $T$ is a homogeneous quartic whose zeros in ${\bf P}^1$ have tetrahedral symmetry, such as $T(X,Y) = X^4 + XY^3$; the surface has $64$ lines, whereas Fermat's $D(X,Y) = D(X',Y')$ has only $48$, with $D$ being a quartic such as $X^4-Y^4$ whose roots have $8$-element dihedral symmetry.] | |
| Jan 3, 2014 at 13:56 | comment | added | jmc | Beauville gave a talk on surfaces with maximal Picard number a few weeks ago: math.ru.nl/~bmoonen/WorkshopDec2013.html (third talk on Tuesday). I am sorry that I did not take notes. | |
| Jan 2, 2014 at 20:54 | comment | added | Alex Degtyarev | Thank you for the answers and for the ref to Beauville. From the latter I conclude that the answer is not known, nor even close. Just to clarify: I'm interested in the maximal Picard number of a nonsingular surface of degree $d$ in $\mathbb{C}P^3$, or at least in its asymptotic. The question is out of pure curiosity (as most math questions are), as the upper and lower bounds that I know differ too much ($2/3d^3$ vs. $3d^2$)! | |
| Jan 2, 2014 at 19:49 | comment | added | Francesco Polizzi | Why not? It is the minimal desingularization of a singular surface (with four $A_9$ singularities). This is a usual construction in order to produce examples with large Picard number, since exceptional curves in the resolution tend to be independent in the Neron-Severi group. | |
| Jan 2, 2014 at 17:39 | comment | added | abx | This quintic example is very nice, but not really smooth... | |
| Jan 2, 2014 at 14:29 | comment | added | Francesco Polizzi | For quintic surfaces, an example with maximum Picard number $\rho = 45$ can be found at arxiv.org/pdf/0812.3519.pdf | |
| Jan 2, 2014 at 13:33 | comment | added | Jason Starr | Just to clarify, are you working in characteristic $0$? In positive characteristic, there are Shioda's unirational surfaces, which I believe are all supersingular. | |
| Jan 2, 2014 at 13:30 | comment | added | Jason Starr | I believe that already for smooth quartic surfaces, Segre found surfaces that beat the Fermat surface. | |
| Jan 2, 2014 at 12:23 | comment | added | Benjamin Dickman | Probably you have already read: math.unice.fr/~beauvill/pubs/Picmax.pdf ? | |
| Jan 2, 2014 at 11:53 | review | First posts | |||
| Jan 2, 2014 at 11:58 | |||||
| Jan 2, 2014 at 11:34 | history | asked | Alex Degtyarev | CC BY-SA 3.0 |