Improving the bounds of the Multiplicity Conjecture: the codimension 3 level case

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra $A$ in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of $A$. All the examples studied so far have lead to conjecture (see $[HZ]$ and $[MNR2]$) that, moreover, the bounds of the MC are sharp if and only if $A$ has a pure MFR. Therefore, it seems a reasonable - and useful - idea to seek better, if possibly {\it ad hoc}, bounds for particular classes of Cohen-Macaulay algebras. In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose $h$-vector is that of a compressed algebra. In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra $A$, which can be expressed exclusively in terms of the $h$-vector of $A$, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of $A$ is not pure. Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras $A$: namely, when $A$ is compressed, or when its $h$-vector $h(A)$ ends with $(...,3,2)$. Also, we will prove our lower bound when $h(A)$ begins with $(1,3,h_2,...)$, where $h_2\leq 4$, and our upper bound when $h(A)$ ends with $(...,h_{c-1},h_c)$, where $h_{c-1}\leq h_c+1$.