Approximating curves on real rational surfaces (Q2804214)

In this paper a real algebraic variety is a classical algebraic variety defined over the reals. If \(X\) is a real algebraic variety, then \(X(\mathbb C)\) denotes the set of its complex points and \(X(\mathbb R)\) the set of its real points. This beautiful article shows (once more) that the behaviour of a real algebraic variety at its complex points plays a fundamental role to understand its set of real points.NEWLINENEWLINEI\textit{J. Bochnak} and \textit{W. Kucharz} [Math. Ann. 314, No. 4, 601--612 (1999; Zbl 0945.14032)] proved that every \({\mathcal C}^{\infty}\) map \(\mathbb S^1\to X\), where \(\mathbb S^1\) denotes the circle and \(X\) is a rational variety, can be approximated in the \({\mathcal C}^{\infty}\)-topology by real algebraic maps \(\mathbb R\mathbb P^1\to X\). The article under review can be seen as the study of a variant of the quoted result. Namely, given a smooth real algebraic variety \(X\) and a smooth, simple, closed curve \(L\subset X(\mathbb R)\), under what conditions can \(L\) be approximated in the \({\mathcal C}^{\infty}\)-topology by the set \(C(\mathbb R)\) of real points of a smooth rational curve \(C\subset X\)? Notice that the smoothness of the varieties \(X, L\) and \(C\) in the question above is not only required for their sets of real points. Indeed, the authors introduce the notion of real-smooth curve to denote a \(1\)-dimensional real algebraic variety \(C\) whose set \(C(\mathbb R)\) of real points is a \(1\)-dimensional real manifold. In other words, \(C(\mathbb C)\) may have singular points but they come in complex conjugate pairs.NEWLINENEWLINEThe main result in the paper is the following.NEWLINENEWLINETheorem. Let \(S(\mathbb R)\) be the set of real points of a smooth rational surface \(S\) and let \(C\subset S(\mathbb R)\) be a connected, closed, \(1\)-dimensional \({\mathcal C}^{\infty}\)-submanifold. The following statements are equivalent:NEWLINENEWLINE(1) \(C\) can be approximated by real-smooth rational curves in the \({\mathcal C}^{\infty}\)-topology.NEWLINENEWLINE(2) There exists a smooth rational surface \(S_1\) and a smooth rational curve \(C_1\subset S_1\) such that the pair \((S(\mathbb R), C)\) is diffeomorphic to the pair \((S_1(\mathbb R), C_1(\mathbb R))\).NEWLINENEWLINE(3) The pair \((S(\mathbb R), C)\) is \textbf{not} diffeomorphic to the pair \((\mathbb T,\mathbb N)\), where \(\mathbb T\) is the \(2\)-torus and \(\mathbb N\) denotes any null homotopic curve.NEWLINENEWLINEThe proof of \((3)\Longrightarrow (2)\) is really involved and requires enumerating all possible topological pairs \((S(\mathbb R),C)\).NEWLINENEWLINETo finish, the authors propose the following conjecture. To state it recall that a variety \(X\) is said to be rationally connected if for two general points \(x,y\) of \(X\) there exists a rational curve \(\Gamma\subset X\) connecting \(x\) and \(y\).NEWLINENEWLINEConjecture. Let \(X\) be a smooth, rationally connected variety defined over \(\mathbb R\). Then a \({\mathcal C}^{\infty}\) map \(\mathbb S^1\to X(\mathbb R)\) can be approximated by rational curves if and only if it is homotopic to a rational curve \(\mathbb R\mathbb P^1\to X(\mathbb R)\).