Cohomology of real three-dimensional triquadrics (Q2880629)

The author considers three quadratic forms \(q_1,q_2, q_3\in \mathbb{R}[X_0,\dots, X_r]\) that define a 3-dimensional nonsingular variety \(V\subseteq\mathbb{C}\mathbb{P}^6\), which is called a triquadric. The matrix of \(q_i\) is denoted by \(M_i\). The spectral curve \(C\subseteq\mathbb{C}\mathbb{P}^2\) of \(V\) is the zero set of the polynomial \(\text{det}(\lambda_1\cdot M_1+ \lambda_2\cdot M_2+ \lambda_3\cdot M_3)\). It is assumed that \(C\) is nonsingular. Then \(\mathbb{R} C\subseteq\mathbb{R}\mathbb{P}^2\) is a nonsingular curve of degree 7.NEWLINENEWLINE The author shows how information about the disposition and the properties of the ovals of \(\mathbb{R} C\) can be used to compute the cohomology (modulo 2) of the real part of the triquadric. The main results consist of a large collection of different cases. It is shown that every case is actually realized by a suitable triquadric.