Let $K$ be a field, $S=K[x_1,\ldots ,x_m, y_1,\ldots , y_n]$ a standard bigraded polynomial ring, and $M$ a finitely generated bigraded $S$-module. In this paper, we study the sequentially Cohen-Macaulayness of~$M$ with respect to $Q=(y_1,\ldots ,y_n)$. We characterize the sequentially Cohen-Macaulayness of $L\otimes _KN$ with respect to $Q$ as an $S$-~module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \ldots , x_m]$ and $K[y_1, \ldots , y_n]$, respectively. All hypersurface rings that are sequentially Cohen-Macaulay with respect to $Q$ are classified.
"Sequentially Cohen-Macaulayness of bigraded modules." Rocky Mountain J. Math. 47 (2) 621 - 635, 2017. https://doi.org/10.1216/RMJ-2017-47-2-621