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The minimal number of generators of a Togliatti system

2016; Springer Science+Business Media; Volume: 195; Issue: 6 Linguagem: Inglês

10.1007/s10231-016-0554-y

1618-1891

Emilia Mezzetti, Rosa M. Miró-Roig,

Advanced Combinatorial Mathematics

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 $$d\ge 2$$ and $$n\ge 2$$ . 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 $$n\ge 3$$ (resp. $$n\ge 4$$ ) 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.