Zero cycles in \(\mathbb{P}^n\).

*(English)*Zbl 0787.14004This paper naturally breaks up into two sections, the first of which is essentially elementary. The second is more technical, and involves flatness. It is well known that if \(K\) is any field and \(K[x_ 1\dots x_ r]\) is the polynomial ring in \(r\) independent variables, the symmetric group can be made to act on the ring by permuting the variables, and the invariant subring is generated by the so-called elementary symmetric functions. In the first section, we study the following generalization. Let \(K\) be a field, \(R = K[X_ 1 \dots X_ r]\), where now the \(X_ i\)’s are vectors of independent variables, \(X_ i = (x_{i1},\dots,x_{in})\). The symmetric group \(\Sigma_ r\) acts on this ring, this time by permuting the vectors. The question arises: what can we say about generators for the invariant subring? It turns out that there is a natural higher-dimensional analogue of the elementary symmetric functions, a finite set of polynomials that we call the multilinear symmetric polynomials. So we ask whether these generate \(R^{\Sigma_ r}\), the invariant subring of \(R\).

The major results of section 1 show that if \(K\) has characteristic zero, the answer is yes. If \(K\) has characteristic \(p\), the answer is no, provided \(n\) and \(r\) are large enough (in particular, we show this if \(n\geq p + 1\) and \(r\geq p +1\)). - It turns out that this result has some significance in relation to varieties parametrizing cycles of \(r\) points in \(\mathbb{P}^ n\). The most natural is the symmetric product, \(\text{Symm}^ r\mathbb{F}^ n\). Another standard one is \(\text{Chow}(r,n)\) – the Chow variety of zero cycles of degree \(r\) in \(\mathbb{P}^ n\). It turns out that the two varieties are isomorphic precisely when the multilinear symmetric polynomials generate \(R^{\Sigma_ r}\).

So far, we have only considered zero cycles – that is, given a zero- dimensional subscheme in \(\mathbb{P}^ n\) we only consider the multiplicities of the points, not the ideal of their embeddings. There is, however, a variety parametrizing zero-dimensional subschemes in \(\mathbb{P}^ n\), namely the Hilbert scheme of subschemes of \(\mathbb{P}^ n\) with Hilbert polynomial \(r\) (the constant). We denote it by \(\text{Hilb}^ r_{\mathbb{P}_ n}\). In section 2 we study this variety, and show that there is a morphism \(\text{Hilb}^ r_{\mathbb{P}_ n}\to \text{Symm}^ r\mathbb{P}^ n\). This is a strengthening of what was already known: J. Fogarty [J. Reine Angew. Math. 234, 65-88 (1969; Zbl 0197.17101)] has shown that there is a morphism \(\text{Hilb}^ r_{\mathbb{P}_ n} \to \text{Chow}(r,n)\).

The major results of section 1 show that if \(K\) has characteristic zero, the answer is yes. If \(K\) has characteristic \(p\), the answer is no, provided \(n\) and \(r\) are large enough (in particular, we show this if \(n\geq p + 1\) and \(r\geq p +1\)). - It turns out that this result has some significance in relation to varieties parametrizing cycles of \(r\) points in \(\mathbb{P}^ n\). The most natural is the symmetric product, \(\text{Symm}^ r\mathbb{F}^ n\). Another standard one is \(\text{Chow}(r,n)\) – the Chow variety of zero cycles of degree \(r\) in \(\mathbb{P}^ n\). It turns out that the two varieties are isomorphic precisely when the multilinear symmetric polynomials generate \(R^{\Sigma_ r}\).

So far, we have only considered zero cycles – that is, given a zero- dimensional subscheme in \(\mathbb{P}^ n\) we only consider the multiplicities of the points, not the ideal of their embeddings. There is, however, a variety parametrizing zero-dimensional subschemes in \(\mathbb{P}^ n\), namely the Hilbert scheme of subschemes of \(\mathbb{P}^ n\) with Hilbert polynomial \(r\) (the constant). We denote it by \(\text{Hilb}^ r_{\mathbb{P}_ n}\). In section 2 we study this variety, and show that there is a morphism \(\text{Hilb}^ r_{\mathbb{P}_ n}\to \text{Symm}^ r\mathbb{P}^ n\). This is a strengthening of what was already known: J. Fogarty [J. Reine Angew. Math. 234, 65-88 (1969; Zbl 0197.17101)] has shown that there is a morphism \(\text{Hilb}^ r_{\mathbb{P}_ n} \to \text{Chow}(r,n)\).

##### Keywords:

multilinear symmetric polynomials; characteristic \(p\); symmetric product; Chow variety; zero cycles; Hilbert scheme
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##### References:

[1] | Fogarty, J, Truncated Hilbert functors, J. reine angew. math., 234, 66-88, (1969) · Zbl 0197.17101 |

[2] | \scA. Grothendieck, “Eléments de Géométrie Algébrique,” Inst. Hautes Études Sci. [With J. Dieudonné] · Zbl 0203.23301 |

[3] | Mumford, D, Geometric invariant theory, () · Zbl 0147.39304 |

[4] | Naganta, M, On the normality of the Chow variety of positive O-cycles of degree in an algebraic variety, Mem. college sci. univ. Kyoto, 29, 165-176, (1955) |

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