The paper is devoted to algebraic surfaces which can be obtained using a simple combinatorial procedure called the T-construction. The class of T-surfaces is sufficiently rich: for example, we construct T-surfaces of an arbitrary degree in RP³ which are M-surfaces. We also present a construction of T-surfaces in RP³ with dim H1 (RX; Z/2) > h1, 1(CX), where RX and CX are the real and the complex point sets of the surface.
In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V → k be the corresponding standard part map. Under a mild assumption on (R,V) we show that a definable set X ⊆ Vⁿ admits a triangulation that induces a triangulation of its standard part st X ⊆ kⁿ.
It follows from the known restrictions on the topology of a real algebraic variety that the number of handles of the real part of a real nonsingular sextic in CP3 is at most 47. We construct a real nonsingular sextic X6 in CP3 whose real part RX6 has 44 handles. In particular, this surface verifies b1(RX6) = h1,1(X6) + 2. We extend the construction in order to obtain for any even m ≥ 6 a real nonsingular surface Xm of degree m in CP3 verifying b1(RXm) > h1,1(Xm). It is known that such a surface...
Let C ⊆ Pn be an unramified nonspecial real space curve having many real branches and few ovals. We show that C is a rational normal curve if n is even, and that C is an M-curve having no ovals if n is odd.
On montre que l’ensemble des matrices tridiagonales périodiques symétriques de spectre fixé possède une direction tangente privilégiée, construite à l’aide des vecteurs propres des matrices et de la jacobienne d’une courbe hyperelliptique. Il se trouve que cette direction est celle du célèbre flot de Toda périodique.
We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at real and pairs of conjugate imaginary points, where , and the real quadric blown up at pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil’s recursive formula [22] for Gromov–Witten invariants of these surfaces and generalizes our recursive formula [12] for purely real Welschinger invariants of real toric...
The aim of this paper is to prove that every subset of definable from addition, multiplication and exponentiation admits a stratification satisfying Whitney’s conditions a) and b).
A compact semialgebraic set admits a semialgebraic triangulation such that the family of open simplexes forms a Whitney stratification and is compatible with a finite number of given semialgebraic subsets.
Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a function f:Rⁿ → R. This implies approximation of definable continuous functions and gluing of functions defined on closed definable sets.