## Toric Hilbert schemes.(English)Zbl 1067.14005

The classical Hilbert scheme introduced by A. Grothendieck [Sém. Bourbaki 1960/1961, Exp. 221 (1961; Zbl 0236.14003)], parametrizes subschemes of $$\mathbb P^r_k, k$$ a field, with a given Hilbert polynomial.
A toric variety is a variety parametrized by a finite number of monomials $${\mathbf t}^{a_i} = t_1^{a_1}\cdots t_d^{a_d}, i = 1,\ldots,n,$$ in the polynomial ring $$k[t_1,\ldots,t_d],$$ where $${\mathcal A} = \{a_1,\ldots,a_n\}$$ denotes a subset of $$\mathbb N^d \setminus 0$$ of $$n$$ different vectors. Let $$S = k[x_1,\ldots, x_n]$$ denote the polynomial ring in the variables $$x_1,\ldots,x_n$$ of degree $$a_1,\ldots, a_n$$ respectively. The toric ideal $$I_{\mathcal A}$$ is the kernel of the natural map $$S \to k[t_1,\ldots, t_d], x_i \mapsto {\mathbf t}^{a_i}, i = 1,\ldots, n,$$ which is a prime $$\mathbb N^d$$-graded ideal.
A homogeneous ideal $$M \subset S$$ is called $$\mathcal A$$-graded if $$\dim_k (S/M)_b = 1$$ if $$b \in \mathbb N \mathcal A$$ and $$0$$ otherwise. That is, $$S/M$$ has the same multigraded Hilbert function as the toric ring $$S/I_{\mathcal A}.$$ The authors construct the toric Hilbert scheme $$H_{\mathcal A}$$ that parametrizes all ideals with the same multigraded Hilbert function as $$S/I_{\mathcal A},$$ satisfying a universality property. It follows that there exists exactly one component containing the point $$[I_{\mathcal A}].$$ If char$$(k)= 0,$$ then this component is reduced and so the point $$[I_{\mathcal A}]$$ on $$H_{\mathcal A}$$ is smooth.
Moreover, in the case of codim$$(S/I_{\mathcal A}) = 2$$ the authors prove the following additional results:
(1) The toric Hilbert scheme has one component. It is the closure of the orbit of the toric ideal under the torus action.
(2) The toric Hilbert scheme is 2-dimensional and smooth. Note that there is no restriction on the characteristic of the field $$k.$$
(3) $$H_{\mathcal A}$$ is the toric variety of the Gröbner fan of $$I_{\mathcal A}.$$
In an unpublished paper, B. Sturmfels started with a different construction in order to parametrize all ideals with the same Hilbert function as $$I_{\mathcal A}$$ [“The geometry of $$\mathcal A$$-graded algebras”, preprint, http://arxiv.org/abs/math.AG/94100032] .

### MSC:

 14C05 Parametrization (Chow and Hilbert schemes) 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14M25 Toric varieties, Newton polyhedra, Okounkov bodies

Zbl 0236.14003

Macaulay2
Full Text:

### References:

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