Mirror configurations of points and lines and algebraic surfaces of degree four
Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part 8, Tome 231 (1995), pp. 299-308
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We prove that mirror nonsingular configurations of m points and $n$ lines in $\mathbb RP^3$ exist only for $m\le3$, $n\equiv0$ or $1\pmod4$ and for $m=0$ or $1\pmod4$, $n\equiv0\pmod2$. In addition, we give an elementary proof of V. M. Kharlamov's well-known result saying that if a nonsingular surface of degree four in $\mathbb RP^3$ is noncontractible and has $M\ge5$ components, then it is nonmirror. For the cases $M=5, 6,7$ and $8$, Kharlamov suggested an elementary proof using an analogy between such surfaces and configurations of $M-1$ points and a line. Our proof covers the remaining cases $M=9,10$. Bibl. 5 titles.
@article{ZNSL_1995_231_a20,
author = {S. S. Podkorytov},
title = {Mirror configurations of points and lines and algebraic surfaces of degree four},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {299--308},
publisher = {mathdoc},
volume = {231},
year = {1995},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_231_a20/}
}
S. S. Podkorytov. Mirror configurations of points and lines and algebraic surfaces of degree four. Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part 8, Tome 231 (1995), pp. 299-308. http://geodesic.mathdoc.fr/item/ZNSL_1995_231_a20/