Models of curves and valuations
Models of curves and valuations
复制标题
曲线和估值模型
DOI：
10.18725/oparu-3275
复制
发表时间：
2015
影响因子：
0.9
通讯作者：
J. Rüth
中科院分区：
文献类型：
--
作者：
J. Rüth
Deligne和Mumford[10]的半稳定约简定理指出，在一个具有离散值的域上，任何绝对不可约的光滑投影曲线都有一个半稳定模型，至少如果一个人允许基域的有限可分扩展。最初的证明并没有提供一个结构，可以在实际中用来计算曲线的半稳定模型。Arzdorf和Wewers最近对半稳定约简定理的证明使用了更容易用于算法构造的技术。为了证明，他们认为曲线是投影线的Y→X = P。对于任何投影线的半稳定模型，他们考虑它在Y中的归一化，Y的正规模型。他们表明，从X的任何模型开始，人们可以找到模型的放大，使得归一化的特殊纤维的奇异性得到改善。这种改进是通过附加在奇异点上的数值不变量来衡量的，它表明该过程在Y的半稳定模型中经过有限多步后终止。本工作明确了其方法的几个方面，并在计算机代数系统中实现了它们。我们将模型表示为离散值的有限集，这些值对应于特殊光纤的不可约分量。Mac Lane[24,25]的理论提供了这种估值的紧凑表示，非常适合算法考虑。特别是，它允许我们有效地计算Y中X模型的归一化，即与归一化相对应的估值。由此，我们也可以立即推断出模型是否具有减少的特殊纤维，这是模型半稳定的必要条件。如果一个模型的特殊光纤不被约简，那么Epp[11]定理保证了基场存在一个有限的扩展，使得它被约简。由于Epp的方法不是完全建设性的，我们讨论了他的方法的替代方案。利用Mac Lane估值理论，给出了在混合特征（0,p）下构造基域扩展的新算法。当p不除Y→x的度数时，我们的算法可以证明是正确的。假设特种光纤被缩减，我们提供了一种新的算法来计算特种光纤的某些仿射斑块的方程。最后，我们在几个例子中通过计算半稳定模型来说明在这项工作中开发的所有技术。
The Semistable Reduction Theorem by Deligne and Mumford [10] states that any absolutely irreducible smooth projective curve over a field with a discrete valuation has a semistable model, at least if one admits a finite separable extension of the base field. The original proof does not provide a construction which can in practice be exploited to compute semistable models of curves. A recent proof of the Semistable Reduction Theorem by Arzdorf and Wewers [2] uses techniques which are more accessible to algorithmic constructions. For their proof they consider a curve as a cover Y → X = P of the projective line. To any semistable model of the projective line they consider its normalization in Y , a normal model of Y . They show that, starting from any model of X, one can find a blowup of the model such that the singularities of the special fibers of the normalizations have improved. The improvement is measured by numeric invariants which are attached to the singularities and which show that the process terminates after finitely many steps with a semistable model of Y . The present work makes several aspects of their approach explicit and realizes them in a computer algebra system. We represent models as finite sets of discrete valuations which correspond to the irreducible components of the special fiber. A theory by Mac Lane [24,25] provides a compact representation of such valuations which is well suited for algorithmic considerations. In particular, it allows us to efficiently compute the normalization of a model of X in Y , i.e., the valuations which correspond to the normalization. From this we can also immediately deduce whether a model has reduced special fiber or not, a necessary condition for a model to be semistable. If the special fiber of a model is not reduced, then a theorem of Epp [11] guarantees that there is a finite extension of the base field which makes it reduced. As Epp’s method is not fully constructive, we discuss alternatives to his approach. Using the theory of Mac Lane valuations, we provide a new algorithm to construct such an extension of the base field when working in mixed characteristic (0, p). Our algorithm is provably correct if p does not divide the degree of Y → X. Assuming that the special fiber is reduced, we provide a new algorithm to compute equations for certain affine patches of the special fiber. Finally, we illustrate all the techniques developed in this work by computing semistable models in several examples.