We give a general upper bound for the arithmetical rank of the ideals generated by the 2-minors of scroll matrices with entries in an arbitrary commutative unit ring.

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.

When a cone is added to a simplicial complex Δ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley–Reisner ideals of the original simplicial complex and the new simplicial complex Δ′. In particular, we show that the arithmetical rank of the Stanley–Reisner ideal of Δ′ equals the projective dimension of the Stanley–Reisner ring of Δ′ if the corresponding equality holds for Δ.

We show that a rational normal scroll can in general be set-theoretically defined by a proper subset of the 2-minors of the associated two-row matrix. This allows us to find a class of rational normal scrolls that are almost set-theoretic complete intersections.

We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, which were later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.

We show that the Stanley–Reisner ideal of the one-dimensional simplicial complex whose diagram is an n-gon is always a set-theoretic complete intersection in any positive characteristic.

The definition of mixed jets includes the finite sequences of vertical vectors tangent to jet bundles. This allows to define differential operators on vertical forms on jet bundles by using mixed jets prolongations. The total exterior differential is a special case.