The minimal number of generators of a Togliatti system

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree d in n + 1 variables, for any d ≥ 2 and n ≥ 2. We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if n = 2 (resp n = 2, 3) all range between the lower and upper bound is covered, while if n ≥ 3 (resp. n ≥ 4) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for n = 2 the Mumford–Takemoto stability of the syzygy bundle associated with smooth monomial Togliatti systems.