On cubic hypersurfaces with vanishing Hessian (Q479279)

This is a very interesting paper. Continuing in the direction of the classification of the homogeneous polynomials satisfying the homogeneous (real) Monge-Ampere equation (but in the complex number field) of dimension \(\leq 5\) (or projective dimension \(\leq 4\)) by the algebraic geometry method in [\textit{A. Garbagnati} and \textit{F. Repetto}, Collect. Math. 60, No. 1, 27--41 (2009; Zbl 1180.14045)], the authors classified the degree \(3\) solutions in dimensions \(6\) and \(7\) (the projective dimensions \(5\) and \(6\)). \textit{O. Hesse} [J. Reine Angew. Math. 42, 117--124 (1851; Zbl 02750640); ibid. 56, 263--269 (1859; Zbl 02750296)] claimed that all the homogeneous polynomial solutions \(f(x_0 , x_1 , \dots , x_n )\) of the homogeneous Monge-Ampere equation \(\det (D^2 f)=0\) is a cone, i.e., after a projective transformation, \(f\) is only a polynomial of \(x_1 , \dots , x_n\). \textit{P. Gordan} and \textit{M. Nöther} [Math. Ann. 10, 547--568 (1876; JFM 08.0064.05)] proved that for \(n\leq 3\), this is true. But it is not true for \(n=4\), e.g., with \(f=x_0 x_4 ^2 +x_1 x_3 ^2 +x_2 x_3 x_4\), and \(f\) is either a cone or the one given by this special \(f\). Notice that \(f=0\) gives a singular hypersurface \(X\) in the projective space \({\mathbb C}\mathbb{P}^n\). [Zbl 1180.14045] gave an modern algebraic geometry proof of this result. In the paper under review, the authors concentrate on the cases that the degree of the homogeneous polynomial is \(3\), i.e., \(X_f\) is a cubic hypersurface in \({\mathbb C}\mathbb{P}^n\). They obtain the classifications for the cases \(n=5, 6\). They prove that the noncone hypersurfaces are the so called special Perazzo cubic hypersurfaces which are generic \({\mathbb C}\mathbb{P}^k\) fiber bundles through the Perazzo map (see Definition 2.9, Theorem 5.3, 5.4, 5.7). The main method is considering the Legendre map \({\mathbb C}\mathbb{P}^n -\rightarrow ({\mathbb C}\mathbb{P}^n )^*\) (or Gauss map; polar map) \[ [x_0, x_1 , \dots , x_n ] \rightarrow [f_0 , f_1 , \dots , f_n ]. \] Since \(f\) satisfies the Monge-Ampere equation, \(f_0 , f_1, \dots , f_n\) are dependent. That is, the image subvariety \(Z\) is only a subset of the projective space. In page 782 after Definition 1.5, the authors define the singularity of this map, which is the subvariety defined by the quadratics \(f_0, f_1, \dots , f_n\). In the point of view of the reviewer, this subvariety should be denoted by \(\text{Sing} f\) but not \(\text{Sing} X\). It is not the singularity of \(X\) and the later notation is very confusing. One major result of [JFM 08.0064.05] is iii) of the corollary 1.10 that \(\text{Sing} f\) is nonempty. Therefore, the standard form for \(f\) could be \(x_0 (\sum _{i=k>0}^n x_i^2) +g(x_1 , \dots x_n )\). If \(n\leq 5\), by checking the \(x_0\) terms in the determinant and some elementary methods, one could prove that \(k=n\). Then the classification. If \(n=6\), one could prove that \(k=5\) or \(6\). When \(k=5\), one get Theorem 5.7, and when \(k=6\) one has Theorem 5.4. For the reviewer, it is very striking that if one replaces \(x_0\) by \(t\) as in [Math. Res. Lett. 6, No. 5--6, 547--555 (1999; Zbl 0968.53050)] through the partial Legendre transformation on \(x_1 , \dots , x_n\), one could get a map between \(f(t, x_1 , \dots , t_n ) \rightarrow F(t, y_1 , \dots , y_n )\) with \(f_{tt} =F_{tt}=0\) and, both \(f\) and \(F\) satisfying the Monge-Ampere equation. One might be wondering how this might help even in the case in which \(f\) is a homogeneous cubic.