On set-theoretic intersections and connectedness of algebraic varieties in \(\mathbb P^n\)
The aim of this note is to show that the use of connectedness does not help so much in estimating the minimum number \(\text{ara}(V)\) of equations which are needed to define a projective variety \(V\) set-theoretically, even in the case that \(V\) is defined by a monomial ideal. It is known that if \(V\subset\mathbb P^n_k\) is connected in codimension \(k\), but not in codimension \(k-1\), and if \(\text{ara}(V)<n\), then \(ara(V)\leq \text{codim} (V)+k-1\) [see e.g. \textit{J. Rung}, ``Mengentheoretische Durchschnitte und Zusammenhang'', Regensburger Math. Schr. 3 (1979; Zbl 0512.14030)]. This note gives a class of examples where \(V\) is defined by a monomial ideal and the difference \(\text{ara}(V)-(\text{codim} (V)+k-1)\) can be arbitrarily large.
scientific article; zbMATH DE number 4059581
scientific article