Toric degenerations of low-degree hypersurfaces. (English) Zbl 1534.14052
In this article, the question is asked, when a hypersurface of degree \(d\) in \(\mathbb P^n\) admits a degeneration to a toric variety.
More precisely, given a weight vector \(w\) order on the homogeneous coordinate ring of \(\mathbb P^n\), a Gröbner degeneration of a hypersurface \(\{f(x_0,\ldots,x_n)=0\}\) is the degeneration to the zero locus of the initial ideal \(\{\mathrm{In}_w(f)=0\}\), where \(\mathrm{In}_w(f)\) consists of all monomials in \(f\) of minimal weight with respect to \(w\). If \(\mathrm{In}_w(f)\) happens to consist of only two monomials, the associated zero locus turns out to be a toric variety.
In the article is shown that if the hypersurface is general, then a Gröbner degeneration to a toric variety exists (up to change of coordinates) if and only if \(d < 2n\).
More precisely, given a weight vector \(w\) order on the homogeneous coordinate ring of \(\mathbb P^n\), a Gröbner degeneration of a hypersurface \(\{f(x_0,\ldots,x_n)=0\}\) is the degeneration to the zero locus of the initial ideal \(\{\mathrm{In}_w(f)=0\}\), where \(\mathrm{In}_w(f)\) consists of all monomials in \(f\) of minimal weight with respect to \(w\). If \(\mathrm{In}_w(f)\) happens to consist of only two monomials, the associated zero locus turns out to be a toric variety.
In the article is shown that if the hypersurface is general, then a Gröbner degeneration to a toric variety exists (up to change of coordinates) if and only if \(d < 2n\).
Reviewer: Andreas Hochenegger (Milano)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 14J70 | Hypersurfaces and algebraic geometry |
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