Rokhlin’s formula for dividing T-curves.

*(English)*Zbl 1075.14054Consider a nonsingular dividing real algebraic plane projective curve (dividing means that the real point set of the curve divides its complex point set into two halves exchanged by the involution of complex conjugation). Rokhlin’s formula provides a relation between the topology of two embeddings of the real point set of the curve: the embedding in the real projective plane and the embedding in the complex point set of the curve. Some generalizations of Rokhlin’s formula can be found in the survey [A. Degtyarev and V. Kharlamov, Russ. Math. Surv. 55, No. 4, 735–814 (2000; Zbl 1014.14030)].

The article under review treats the case of \(T\)-curves, i.e., the curves which can be obtained using the combinatorial patchworking. This construction is a particular case of the Viro method of construction of real algebraic varieties with prescribed topology. Let \(m\) be a positive integer, and \(T_m\) the triangle with vertices \((0,0)\), \((m,0)\), \((0,m)\). Starting with a convex triangulation (sometimes these triangulations are also called regular or coherent) with integer vertices of \(T_m\) and a distribution of signs at the vertices of the triangulation, the combinatorial patchworking produces a nonsingular curve (called a \(T\)-curve) of degree \(m\) in the real projective plane (for details, see, for example, [I. Itenberg and O. Viro, Math. Intell. 18, No. 4, 19–28 (1996; Zbl 0876.14017)]).

The article under review contains a purely combinatorial proof of Rokhlin’s formula in the case of dividing \(T\)-curves obtained from primitive triangulations (i.e., triangulations in triangles of area 1/2). The proof is based on a combinatorial description of dividing \(T\)-curves proposed by the author [Geom. Dedicata 101, 129–151 (2003; Zbl 1069.14062)], and works without the assumption of convexity of the triangulation (the importance of the latter remark can be underlined by the fact that, as it is shown in [I. Itenberg and E. Shustin, Turk. J. Math. 26, No. 1, 27–51 (2002; Zbl 1047.14047)], for non-convex triangulations of \(T_m\) the combinatorial patchworking produces pseudo-holomorphic curves).

The article under review treats the case of \(T\)-curves, i.e., the curves which can be obtained using the combinatorial patchworking. This construction is a particular case of the Viro method of construction of real algebraic varieties with prescribed topology. Let \(m\) be a positive integer, and \(T_m\) the triangle with vertices \((0,0)\), \((m,0)\), \((0,m)\). Starting with a convex triangulation (sometimes these triangulations are also called regular or coherent) with integer vertices of \(T_m\) and a distribution of signs at the vertices of the triangulation, the combinatorial patchworking produces a nonsingular curve (called a \(T\)-curve) of degree \(m\) in the real projective plane (for details, see, for example, [I. Itenberg and O. Viro, Math. Intell. 18, No. 4, 19–28 (1996; Zbl 0876.14017)]).

The article under review contains a purely combinatorial proof of Rokhlin’s formula in the case of dividing \(T\)-curves obtained from primitive triangulations (i.e., triangulations in triangles of area 1/2). The proof is based on a combinatorial description of dividing \(T\)-curves proposed by the author [Geom. Dedicata 101, 129–151 (2003; Zbl 1069.14062)], and works without the assumption of convexity of the triangulation (the importance of the latter remark can be underlined by the fact that, as it is shown in [I. Itenberg and E. Shustin, Turk. J. Math. 26, No. 1, 27–51 (2002; Zbl 1047.14047)], for non-convex triangulations of \(T_m\) the combinatorial patchworking produces pseudo-holomorphic curves).

Reviewer: Ilia Itenberg (Strasbourg)