A linear form, $\ell$, is a Weak Lefschetz element for an Artinian Algebra $A$ if the multiplication map $\times$$\ell$ from $A_{i}$ to $A_{i+1}$ has maximal rank for each integer $i$. The set of linear forms with this property forms a Zariski-open set and its complement is called the non-Lefschetz locus. In fact this locus can be given a scheme structure that is not always reduced.

In this talk, we want to define in a similar way the non-Lefschetz locus for conics. We say that $C$, a homogeneous polynomial of degree $2$, is a Lefschetz conic for $A$ if the multiplication map $\times$$C$ from $A_{i}$ to $A_{i+2}$ has maximal rank for each integer $i$.

An important result by Boij, Migliore, Miró-Roig and Nagel proved that for a general Artinian complete intersection of height $3$, the non-Lefschetz locus has the expected codimension and the expected degree.

This can be generalized to certain finite length modules $M$ which are the first cohomology module of a vector bundle $\mathcal{E}$ of rank $2$ over $\mathbb{P}^{2}$. It has been proven by Failla, Flores and Peterson that these modules have the Weak Lefschetz property. The non-Lefschetz locus, in this case, is exactly the set of jumping lines of $\mathcal{E}$, and the expected codimension is achieved under the assumption that $\mathcal{E}$ is general.

We address the same type of questions for the non-Lefschetz locus of conics. In this case, the non-Lefschetz locus of conics is always a subset of the jumping conics, but these sets do not coincide in the case when $\mathcal{E}$ is semistable with first Chern class even.

Finally, we will show that for general complete intersection of height $3$, the non-Lefschetz locus has the expected codimension as a subscheme of $\mathbb{P}^{5}$, and that the same does not hold for certain monomial complete intersections.