We consider the Hilbert scheme $H(d,g)$ of space curves $C$ with homogeneous ideal $I\left(C\right):=H_{*}^0\left({ℐ}_C\right)$ and Rao module $M:=H_{*}^1\left({ℐ}_C\right)$. By taking suitable generizations (deformations to a more general curve) $C^{\prime }$ of $C$, we simplify the minimal free resolution of $I\left(C\right)$ by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of $I\left(C^{\prime }\right)$. Using this for Buchsbaum curves of diameter one ($M_v\ne 0$ for only one $v$), we establish a one-to-one correspondence between the set $𝒮$ of irreducible components of $H(d,g)$ that contain $\left(C\right)$ and a set of minimal...

This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$, whose main results deal with curves which satisfy ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. of diameter, $\mathrm{diam}M\le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\mathrm{H}(d,g)$, at $\left(C\right)$ under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of $C$ turns out to be equivalent to the...

For each squarefree monomial ideal $I\subset S=k[x_1,...,x_n]$, we associate a simple finite graph $G_I$ by using the first linear syzygies of $I$. The nodes of $G_I$ are the generators of $I$, and two vertices $u_i$ and $u_j$ are adjacent if there exist variables $x,y$ such that $x{u}_i=y{u}_j$. In the cases, where $G_I$ is a cycle or a tree, we show that $I$ has a linear resolution if and only if $I$ has linear quotients and if and only if $I$ is variable-decomposable. In addition, with the same assumption on $G_I$, we characterize all squarefree monomial ideals with a...

Let $G$ be a finite simple graph with the vertex set $V$ and let $I_G$ be its edge ideal in the polynomial ring $S=𝕂\left[V\right]$. We compute the depth and the Castelnuovo-Mumford regularity of $S/I_G$ when $G=G_1\circ G_2$ or $G=G_1*G_2$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_1$, $G_2$ by the $\circ$ operation or $*$ operation, respectively.

We consider a generalization of the notion of torsion theory, which is associated with a Serre subcategory over a commutative Noetherian ring. In 2008 Aghapournahr and Melkersson investigated the question of when local cohomology modules belong to a Serre subcategory of the module category. In their study, the notion of Melkersson condition was defined as a suitable condition in local cohomology theory. One of our purposes in this paper is to show how naturally the concept of Melkersson condition...