- Mateusz Michałek (MPI MiS, Leipzig)
Togliatti systems join in a beutiful way two seemingly unrelated topics:
1) weak Lefschetz property (WLP) and
2) varieties with degenerate osculating spaces.
The first examples were presented already in 1929 by Togliatti who studied monomial maps from P2 to P5. However, only recently Mezzetti, Miró-Roig and Ottaviani, using apolarity, proved general results relating projections of Veronese embedding with degenerate general osculating spaces and Artinian ideals that fail the WLP. In our talk we will present the above mentioned algebraic and geometric properties, providing further examples. We will define Togliatti systems and show new results, obtained jointly with Miró-Roig, on their classification, answering a conjecture of Ilardi (corrected by Mezzetti, Miró-Roig and Ottaviani).