Manin conjecture

In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators[1] in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.

Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface.

Conjecture

Their main conjecture is as follows. Let $V$ be a Fano variety defined over a number field $K$ , let $H$ be a height function which is relative to the anticanonical divisor and assume that $V(K)$ is Zariski dense in $V$ . Then there exists a non-empty Zariski open subset $U\subset V$ such that the counting function of $K$ -rational points of bounded height, defined by

N_{U,H}(B)=\#\{x\in U(K):H(x)\leq B\}

for $B\geq 1$ , satisfies

N_{U,H}(B)\sim cB(\log B)^{\rho -1},

as $B\to \infty .$ Here $\rho$ is the rank of the Picard group of $V$ and $c$ is a positive constant which later received a conjectural interpretation by Peyre.[2]

Manin's conjecture has been decided for special families of varieties,[3] but is still open in general.

gollark: I would probably use nginx, because I'm used to it and it has nicer configuration:```nginxhttp { # whatever important configuration you have for all HTTP servers, `nginx.conf` probably ships with some # fallback in case someone visits with an unrecognized Host header server { listen 80 default_server; listen [::]:80 default_server; return 301 http://somedomain$request_uri; } server { listen 80; # you may (probably do) want HTTPS instead, in which case this bit is somewhat different - you need to deal with certs and stuff, and use port 443 - also you should probably add HTTP/2 listen [::]:80; # IPv6 server_name domain1.com; location / { proxy_pass http://backend1:8080/; } } server { listen 80; listen [::]:80; server_name domain2.com; location / { proxy_pass http://backend2:8080/; } }}```

gollark: The reverse-proxy solution is in my opinion the best one, although it would require some config.

gollark: I think LetsEncrypt may not be very happy with that, though.

gollark: Yes, and you can just use a reverse proxy (with "vhosts" or whatever) for that, easy enough.

gollark: I think those are just what some webservers call "doing different things based on the host header".

References

Franke, J.; Manin, Y. I.; Tschinkel, Y. (1989). "Rational points of bounded height on Fano varieties". Inventiones Mathematicae. 95 (2): 421–435. doi:10.1007/bf01393904. MR 0974910. Zbl 0674.14012.
Peyre, E. (1995). "Hauteurs et mesures de Tamagawa sur les variétés de Fano". Duke Mathematical Journal. 79 (1): 101–218. doi:10.1215/S0012-7094-95-07904-6. MR 1340296. Zbl 0901.14025.
Browning, T. D. (2007). "An overview of Manin's conjecture for del Pezzo surfaces". In Duke, William (ed.). Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005. Analytic number theory, Clay Math. Proc. Clay Mathematics Proceedings. 7. Providence, RI: American Mathematical Society. pp. 39–55. ISBN 978-0-8218-4307-9. MR 2362193. Zbl 1134.14017.

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[1] Franke, J.; Manin, Y. I.; Tschinkel, Y. (1989). "Rational points of bounded height on Fano varieties". Inventiones Mathematicae. 95 (2): 421–435. doi:10.1007/bf01393904. MR 0974910. Zbl 0674.14012.

[2] Peyre, E. (1995). "Hauteurs et mesures de Tamagawa sur les variétés de Fano". Duke Mathematical Journal. 79 (1): 101–218. doi:10.1215/S0012-7094-95-07904-6. MR 1340296. Zbl 0901.14025.

[3] Browning, T. D. (2007). "An overview of Manin's conjecture for del Pezzo surfaces". In Duke, William (ed.). Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005. Analytic number theory, Clay Math. Proc. Clay Mathematics Proceedings. 7. Providence, RI: American Mathematical Society. pp. 39–55. ISBN 978-0-8218-4307-9. MR 2362193. Zbl 1134.14017.