On the projectivity of the moduli spaces of curves.
On the projectivity of the moduli spaces of curves.
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关于曲线模空间的射影性。
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发表时间：
1993
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通讯作者：
M. Cornalba
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作者：
M. Cornalba
1. 基本上有三模的投射心理空间的代数证明稳定的曲线:原来的克努森[8][9],芒福德的证据[12]和Gieseker[5],这使得大量使用的机械几何不变量理论,和最近的一个Viehweg[15]和科勒[10],而不是依赖的semipositivity直接的图像相对dualizing捆的权力,至少在科勒的版本,不使用几何不变量理论。在本文中，结合上述论文和[4]的思想，我们将概述一个投影性的证明，我们认为，它比任何现有的证明都更简单，更直接。然而，尽管Mumford、Gieseker、Viehweg和Kollar的证明至少在原则上适用于各种各样的模问题，但我们的证明以一种基本的方式使用了手头问题的特殊性，很难看出它如何可以扩展到其他情况。Mumford的思想是利用几何不变理论的机制，通过射影线性群的作用，将稳定曲线的模空间直接构造为多元嵌入稳定曲线的Chow或Hilbert格式的射影商。为了能够做到这一点，我们需要证明稳定曲线的k-正则象在不变理论意义上对于高k是稳定的。对于光滑曲线来说，这是很容易理解的，尽管并不容易；注意到k-正则嵌入稳定曲线是线性稳定的，然后证明了对于光滑曲线，线性稳定性意味着不变量理论稳定性。相比之下，在个别情况下，芒福德和吉斯塞克都不得不依靠相当冗长和复杂的间接论证。我们的主要观点是，只要我们愿意承认模空间作为完全代数空间存在，就可以避免证明奇异稳定曲线的不变理论稳定性；顺便说一句，证明这一点相对容易。与[10]一样，我们通过将标准数值振幅准则之一（在我们的例子中是Seshadri准则）应用于合适的可逆层来证明投影性。利用[4]技术得到了必要的数值估计。更确切地说，利用线性稳定光滑曲线的不变理论稳定性证明了稳定曲线族的一个不等式，该不等式的解释表明，对于不包含在边界内的模曲线，Seshadri判据的假设是成立的。然后用标准初等技术将曲线位于边界的情况简化为前一种情况；这一过程可以看作是对属的一种归纳法，为了恰当地实现这一过程，同时处理稳定n点曲线的模空间也很方便。因此，在某种意义上，我们的方法是使用不变理论稳定性作为获得数值不等式的一种手段，而不是作为通过群作用构造商的一个步骤。
1. There are basically three algebraic proofs of the projectivity of the moduli spaces of stable curves: the original one by Knudsen [8][9], the proof by Mumford [12] and Gieseker [5], which makes heavy use of the machinery of geometric invariant theory, and the more recent one by Viehweg [15] and Kollar [10], which relies instead on the semipositivity of the direct images of powers of the relative dualizing sheaf and, at least in Kollar’s version, does not use geometric invariant theory at all. In this note, combining ideas from the above papers and from [4], we shall outline a proof of projectivity which, we believe, is simpler and more direct than any of the existing ones. However, while the proofs by Mumford, Gieseker, Viehweg, and Kollar are applicable, at least in principle, to a wide variety of moduli problems, ours uses in an essential way the peculiarities of the problem at hand, and it is hard to see how it could be extended to other situations. Mumford’s idea is to use the machinery of geometric invariant theory to directly construct the moduli space of stable curves as a projective quotient of the Chow or Hilbert scheme of pluricanonically embedded stable curves by the action of the projective linear group. To be able to do so, one needs to show that the k-canonical images of stable curves are stable in the invariant-theoretic sense for high k. This is quite well understood, although not really easy, for smooth curves; one notices that k-canonically embedded stable curves are linearly stable, and then proves that, for smooth curves, linear stability implies invariant-theoretic stability. In the singular case, by contrast, both Mumford and Gieseker have to rely on indirect arguments which are quite long and involved. Our main point is that one can avoid proving the invariant-theoretic stability of singular stable curves provided one is willing to grant that moduli space exists as a complete algebraic space; that this is the case, incidentally, is relatively easy to prove. As in [10], we prove projectivity by applying to a suitable invertible sheaf one of the standard numerical ampleness criteria (Seshadri’s criterion in our case). The necessary numerical estimates are obtained using the techniques of [4]. More precisely, the invariant-theoretic stability of linearly stable smooth curves is used to prove an inequality for families of stable curves which, suitably interpreted, says that the hypothesis of Seshadri’s criterion is satisfied for curves in moduli which are not contained in the boundary. The case of curves lying in the boundary is then reduced to the previous one by standard elementary techniques; in order to properly carry out this procedure, which can be viewed as a sort of induction on the genus, it is convenient to deal simultaneously with the moduli spaces of stable n-pointed curves as well. In a sense, then, our approach is to use invariant-theoretic stability as a means of obtaining numerical inequalities, rather than as a step in the construction of quotients by group actions.