Irreducible noncommutative quadrics

In this paper, we study irreducible noncommutative quadrics $S/(f)$ via noncommutative graded matrix factorizations. We show that the line modules over $S/(f)$ are described by the rulings arising from indecomposable noncommutative linear matrix factorizations of $f$ of rank $2$. We study when Zhang twists of a standard smooth irreducible noncommutative quadric are standard. Finally, by identifying all singular central Sklyanin quadrics, we prove that every smooth central Sklyanin quadric is standard.