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In this paper we characterize the componentwise lexsegment ideals which are componentwise linear and the lexsegment ideals generated in one degree which are Gotzmann.
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Let $Isubset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
In this paper it is shown that a sortable ideal $I$ is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals with constant bound.
For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal $I$ and any monomial prime ideal $P$, $text{HS}_i(I(P))subseteq text{HS}_i(I)(P)$ for all $i$, where $I(P)$ is the monomial localization of $I$. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any $textbf{c}$-bounded principal Borel ideal $I$ and for the edge ideal of complement of any path graph, it is proved that $text{HS}_i(I)$ has linear quotients for all $i$. As an example of $textbf{c}$-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, $text{HS}_j(I)$ is again a polymatroidal ideal for all $j$. Moreover, for any edge ideal with linear resolution, the ideal $text{HS}_j(I)$ is characterized and it is shown that $text{HS}_1(I)$ has linear quotients.
We show that the ideal generated by maximal minors (i.e., $(k+1)$-minors) of a $(k+1) times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1,...,1)$.
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