Limit linear series: Basic theory

Limit linear series: Basic theory
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极限线性级数：基础理论

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发表时间：
1986
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通讯作者：
J. Harris
J. Harris
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文献类型：
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作者：
D. Eisenbud;J. Harris

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本文介绍光滑曲线上线性级数退化为某种可约曲线，即紧致型曲线时的处理技巧。Beauville [2]、Knudsen [21-23]、Harris和Mumford [17]以“可容许覆盖”的名义发展了技术上更简单的1维级数特例。它对于研究曲线的模空间（上述论文和Harris [16]）和最简单的Weierstrass点（迪亚兹[4]）是非常有用的。通过我们的扩展工具，我们能够证明，例如：1）亏格曲线的模空间Mg具有一般型forg = 24，且具有科代拉维数forg=23，推广和简化了Harris和Mumford [17]及Harris [16]的工作（并且在更一般的情况下）至少存在一个具有半群Γ的Weierstrass点的曲线的Mg的子簇的分量，其具有“期望”维数3g-2−w（具体而言，3）光滑亏格曲线空间的基本群具有不同的“平凡”Weierstrass点作为完全对称群以单值作用于Weierstrass点。4）如果选择fr和d， $$ ho：= g -（r + 1）（g - d + r）= 0，$$ 则genusg的一般曲线具有某个有限数目的gdr [15，20]。我们表明，家庭的所有这些，允许曲线之间的变化一般曲线，是不可约的，所以这个家庭的单值行为传递。如果4 =1，我们进一步证明了单值作为完全对称群。5）如果fr和d被选择，使得 $$ ho = - 1，$$ 则由具有agdr的曲线组成的Mg的子簇恰好有一个余维不可约分量1.6）对于任意r，g，d使得ρ ≠ 0，由具有agdr的曲线组成的Mg的子簇至少有一个余维不可约分量-ρ，只要 $$ ho geqq left{ egin{gathered} - g + r + 3（r odd）hfill \ - frac{r}{{r + 2}}g + r + 3（r even）. hfill \ end{gathered} 八美元 在本文中，我们提出的基本理论“极限线性级数”证明这些结果所必需的。这些结果将在我们即将发表的论文中讨论[8-12]。更简单的应用程序，不需要本文中开发的工具，但可能通过它们来澄清，已经在我们的论文中给出了[5-7]。
AbstractIn this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville [2], Knudsen [21–23], Harris and Mumford [17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris [16]) and the simplest sorts of Weierstrass points (Diaz [4]). With our extended tools we are able to prove, for example, that:1)The Moduli spaceMg of curves of genusg has general type forg≧24, and has Kodaira dimension ≧1 forg=23, extending and simplifying the work of Harris and Mumford [17] and Harris [16].2)Given a Weierstrass semigroup Γ of genusg and weightw≦g/2 (and in a somewhat more general case) there exists at least one component of the subvariety ofMg of curves possessing a Weierstrass point of semigroup Γ which has the “expected” dimension 3g-2−w (and in particular, this set is not empty).3)The fundamental group of the space of smooth genusg curves having distinct “ordinary” Weierstrass points acts on the Weierstrass points by monodromy as the full symmetric group.4)Ifr andd are chosen so that $$ ho : = g - (r + 1)(g - d + r) = 0,$$ then the general curve of genusg has a certain finite number ofgdr’s [15, 20]. We show that the family of all these, allowing the curve to vary among general curves, is irreducible, so that the monodromy of this family acts transitively. If4=1, we show further that the monodromy acts as the full symmetric group.5)Ifr andd are chosen so that $$ ho = - 1,$$ then the subvariety ofMg consisting of curves posessing agdr has exactly one irreducible component of codimension 1.6)For anyr, g, d such that ρ≦0, the subvariety ofMg consisting of curves possessing agdr has at least one irreducible component of codimension—ρ so long as $$ ho geqq left{ egin{gathered} - g + r + 3 (r odd) hfill \ - frac{r}{{r + 2}}g + r + 3 (r even). hfill \ end{gathered} ight.$$ In this paper we present the basic theory of “limit linear series” necessary for proving these results. The results themselves will be taken up in our forthcoming papers [8-12]. Simpler applications, not requiring the tools developed in this paper but perhaps clarified by them, have already been given in our papers [5-7].