Let $k$ be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers ${bf a}=(a_1,a_2,a_3,a_4)$ defines a Gorenstein non complete intersection monomial curve ${mathcal C}({bf a})$ in ${mathbb A}_k^4$, then the
re exist two vectors ${bf u}$ and ${bf v}$ such that ${mathcal C}({bf a}+t{bf u})$ and ${mathcal C}({bf a}+t{bf v})$ are also Gorenstein non complete intersection affine monomial curves for almost all $tgeq 0$.
Let $R = k[w, x_1,..., x_n]/I$ be a graded Gorenstein Artin algebra . Then $I = ann F$ for some $F$ in the divided power algebra $k_{DP}[W, X_1,..., X_n]$. If $RI_2$ is a height one idealgenerated by $n$ quadrics, then $I_2 subset (w)$ after a possib
le change of variables. Let $J = I cap k[x_1,..., x_n]$. Then $mu(I) le mu(J)+n+1$ and $I$ is said to be generic if $mu(I) = mu(J) + n+1$. In this article we prove necessary conditions, in terms of $F$, for an ideal to be generic. With some extra assumptions on the exponents of terms of $F$, we obtain a characterization for $I = ann F$ to be generic in codimension four.
We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.
We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-Soderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples
of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)].
