Togliatti systems associated to the dihedral group and the weak Lefschetz property. (English) Zbl 1523.14079
Summary: In this note, we study Togliatti systems generated by invariants of the dihedral group \(D_{2d}\) acting on \(k[x_0, x_1, x_2]\). This leads to the first family of non-monomial Togliatti systems, which we call GT-systems with group \(D_{2d}\). We study their associated varieties \(S_{D_{2d}}\), called GT-surfaces with group \(D_{2d}\). We prove that there are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, \(I(S_{D_{2d}})\), is minimally generated by quadrics and we find a minimal free resolution of \(I(S_{D_{2d}})\).
MSC:
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13C14 | Cohen-Macaulay modules |
| 14L30 | Group actions on varieties or schemes (quotients) |
Software:
Macaulay2References:
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