On the arithmetical rank of certain Segre embeddings (Q2841396)

Consider the following question: let \(Z=X\times\mathbb{P}^m\subseteq\mathbb{P}^N\) be the Segre product, where \(X\subseteq\mathbb{P}^n\) is a smooth hypersurface of degree \(d\), \(n\geq 2\) and \(N=nm+n+m\). What are the number of defining equations of \(Z\)?NEWLINENEWLINEIn the paper under review, the author obtains, among other results, the following partial answers to the above question.NEWLINENEWLINEFirst of all, suppose that \(n=2\) and \(m=1\), and let \(X=C\) be a smooth projective curve such that there is a point \(P\in C\) at which the intersection multiplicity of \(C\) and the tangent line at \(P\) is equal to \(d\). Then, it is shown (see Corollary 3.5) that the number of defining equations of \(C\times\mathbb{P}^1\) is equal to \(4\) provided \(d\geq 3\); this theorem recovers and extends previous results obtained by \textit{Q. Song} [Questions in local cohomology and tight closure. PhD thesis, Ann Arbor, MI (2007)], \textit{A. K. Singh} and \textit{U. Walther} [Contemp. Math. 390, 147--155 (2005; Zbl 1191.14059)].NEWLINENEWLINESecond, it is also proved that, if \(X\subseteq\mathbb{P}^n\) is a general smooth hypersurface of degree \(d\leq 2n-1\), then the number of defining equations of \(X\times\mathbb{P}^1\) is less or equal than \(2n\) (see Corollary 3.8).NEWLINENEWLINEThirdly, if \(X=C\) is a smooth conic of \(\mathbb{P}^2\) and the characteristic of the base field is different from \(2\), then the number of defining equations of \(C\times\mathbb{P}^m\) is equal to \(3m\) (see Theorem 3.11).NEWLINENEWLINEFinally, for arbitrary \(n,m\) and \(d\), it is proved (see Proposition 2.9) that, if \(X\) is smooth, then then the number of defining equations of \(X\times\mathbb{P}^m\subseteq\mathbb{P}^N\) can only be either \(N-2\), \(N-1\), or \(N\).