Sequentially Cohen--Macaulayness of bigraded modules

Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring and $M$ a finitely generated bigraded $S$-module. In this paper we study sequentially Cohen--Macaulayness of $M$ with respect to $Q=(y_1,\ldots,y_n)$. We characterize the sequentially Cohen--Macaulayness of $L\tensor_KN$ with respect to $Q$ as an $S$-module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \dots, x_m]$ and $K[y_1, \dots, y_n]$, respectively. All hypersurface rings that are sequentially Cohen--Macaulay with respect to $Q$ are classified.