PROPERTIES OF $T$-SPREAD PRINCIPAL BOREL IDEALS GENERATED IN DEGREE TWO

In this paper, we have studied the stability of $t$-spread principal Borel ideals in degree two. We have proved that $\Ass^\infty(I) =\Min(I)\cup \{\mathfrak{m}\}$ , where $I=B_t(u)\subset S$ is a $t$-spread Borel ideal generated in degree $2$ with $u=x_ix_n, t+1\leq i\leq n-t.$ Indeed, $I$ has the property that $\Ass(I^m)=\Ass(I)$ for all $m\geq 1$ and $i\leq t,$ in other words, $I$ is normally torsion free. Moreover, we have shown that $I$ is a set theoretic complete intersection if and only if $u=x_{n-t}x_n$. Also, we have derived some results on the vanishing of Lyubeznik numbers of these ideals.

EQUIVARIANT -MODULES ON ALTERNATING SENARY 3-TENSORS

We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$ . We describe explicitly the category of $\operatorname{GL}_{6}$ -equivariant coherent ${\mathcal{D}}_{X}$ -modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$ -modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$ -structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

Connectedness and Lyubeznik Numbers

We investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals 1. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number $\lambda _{1,2}(A)$ of the local ring $A$ at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.