Mirror configurations of points and lines and algebraic surfaces of degree four

We prove that mirror nonsingular configurations of m points and n lines in ℝP ³ exist only for m≤3, n≡0 or 1 (mod 4) and for m=0 or 1 (mod 4), n≡0 (mod 2). 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 ℝP ³ is noncontractible and has M≥5 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. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 299–308. Translated by N. Yu. Netsvetaev.