ACM curves in multiprojective spaces (Q2087718)

A \textit{multiprojective space} is a product \(Y = \prod_{i=1}^k \mathbb P^{n_i}\) of projective spaces. The Picard group of \(Y\) consists of the line bundles \(\mathcal O_Y (d_1, \dots, d_k) = \otimes \pi_i^*(\mathcal O_{\mathbb P^{n_i}} (d_i))\) with \((d_1, \dots, d_k) \in \mathbb Z^k\). A closed subscheme \(T \subset Y\) is Arithmetically Cohen-Macaulay (ACM) if the restriction maps \(H^0(\mathcal O_Y (b_1, \dots, b_k)) \to H^0(\mathcal O_T (b_1, \dots, b_k))\) are surjective for all \((b_1, \dots, b_k) \in \mathbb N^k\). Extending the well known case \(k=1\), the author proves that if \(T \subset Y\) is ACM and \(\dim T = 1\), then \(T\) is connected with no embedded points; further, if \(\pi_i|_T\) is finite, then \(\pi_i|_T\) is an embedding, in particular \(\pi_i|_T\) is an isomorphism if \(n_i = 1\). The main results are these. If \(T \subset Y\) is an integral, non-degenerate ACM curve of multidegree \((a_1, \dots, a_k)\) and some \(n_i = 1\), then \(T\) is a smooth rational curve and each projection \(\pi_h|_T\) embeds \(T\) as a rational normal curve in \(\mathbb P^{n_h}\). If some \(n_i=2\), then \(T\) is isomorphic to a plane curve of degree \(a_i\). If \(Y = (\mathbb P^1)^k\), then \(T\) is ACM \(\iff T\) is connected and has multidegree \((1,\dots,1) \iff T\) is connected, has multidegree \((1, \dots, 1)\) with irreducible smooth rational components and at worst seminormal singularities and arithmetic genus \(p_a (T) = 0\). For example, in \(\mathbb P^1 \times \mathbb P^1\), take \(T\) to be the union of the coordinate axes. \{ Reviewer's remark: The author defines \(T \subset Y\) to be non-degenerate if (a) there is no multiprojective space \(T \subset Y^\prime \subset Y\) and says this is equivalent to (b) each projection \(\pi_i (T)\) is contained in no hyperplane in \(\mathbb P^{n_i}\). The diagonal inclusion \(\Delta: \mathbb P^n \hookrightarrow \mathbb P^n \times \mathbb P^n\) shows that these two are not equivalent. Looking at the paper, it seems clear that (b) is the definition that the author is using. \}