Topological classification of real three-dimensional cubics
Let \(W=\sum_{|I|=3}w_IX^I\) be a complex cubic form in the variables \(X_0,\dots ,X_4\) with complex coefficients \(w_I\in\mathbb C\), where \(I=(i_0,\dots ,i_4)\in (\mathbb Z_{\geq 0})^5\), \(|I|=\sum_{k=0}^4i_k\) and \(X^I=X_0^{i_0}\dots X_4^{i_4}\). Because the number of the coefficients of \(W\) is \(\binom{3+4}{4}=35\), we denote by \({\mathcal B}\) the complex projective space \(\mathbb P^{35}\) consisting of all homogeneous coordinates \([(w_I)]_{I=(i_0,\dots ,i_4);|I|=4}\) of the corresponding complex cubic forms. For each \(b=[w_I]\in {\mathcal B}=\mathbb P^{35}\), let \(V_b\subset\mathbb P^4\) denote the hypersurface defined by the equation \(\sum_{|I|=3}w_IX^I=0\), and let \({\mathcal B}^*\subset {\mathcal B}\) denote the subspace consisting of all \(b\in {\mathcal B}\) such that \(V_b\) is non-singular. Let \(\mathbb R{\mathcal B}^*\) and \(\mathbb R V_b\) (\(b\in {\mathcal B}^*\)) denote the corresponding \textit{real} points of \({\mathcal B}^*\) and \(V_b\). Previously, the author studied the topology of \(\mathbb R{\mathcal B}^*\) and he proved that there is a following decomposition of connected components: \[ \mathbb R{\mathcal B}^*=\bigg(\bigcup_{k=0}^5{\mathcal B}(k)\bigg) \cup \big({\mathcal B}(0)_I^{\prime}\cup{\mathcal B}(0)_I^{\prime\prime} \cup {\mathcal B}(0)_{II}\big)\cup \big({\mathcal B}(3)_I\cup {\mathcal B}(3)_{II}\big). \tag \(*\) \] In this paper, he determines that all topological types of of non-singular real three dimensional cubics explicitly. In particular, he obtains the explicit topological classification of real cubics for each connected components \((*)\).
scientific article; zbMATH DE number 5628216