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Gröbner techniques for low-degree Hilbert stability. (English) Zbl 1267.14059

Let \(V\) be an \(N+1\)-dimensional \(k\)-vector space, \(k\) an algebraically closed field.
Hilbert stability of bicanonical models of curves of small genus with suitable large automorphism groups with respect to linearizations of fixed small degree is studied. A method is given for deducing the stability (with respect to \(SL(V)\)) of the Hilbert point of a subscheme \(X\) of \(\mathbb{P}(V)\) from a symbolic calculation of certain state polytopes.
The method is implemented in the package StatePolytope in Macaulay2 based on the packages gfan and polymake. Several examples are treated among them the so–called Wiman curves (special hyperelliptic curves) and joins of Wiman curves.

MSC:

14L24 Geometric invariant theory
14H10 Families, moduli of curves (algebraic)
14D22 Fine and coarse moduli spaces
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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References:

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