Examples of Nonsingular Irreducible Curves Which Give Reducible Singular Points of ${\rm red}(H\sb {d,g})$
Examples of Nonsingular Irreducible Curves Which Give Reducible Singular Points of ${\rm red}(H\sb {d,g})$
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给出 ${ m red}(Hsb {d,g})$ 的可约奇异点的非奇异不可约曲线示例
DOI：
10.2977/prims/1195178928
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
M. Amasaki
中科院分区：
文献类型：
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作者：
M. Amasaki
Hilb(P)的开子格式HDig由d次和亏格为g的非奇异不可约曲线对应的点组成，在许多情况下具有多个不可约分支。如果一个人进一步关心它的连通分量，他将不可避免地遇到一个问题，即是否为红色(HDig.)在与给定曲线对应的点上不可约或不可约，但即使是RED(H^)的这种可约奇点的例子似乎也不是很清楚，除了J.Harris在[6；p.93]中提到P“(n^4)中存在非退化的非奇异不可约曲线，其Hilbert点位于Hilb(P”)的多个不可约分支上。这些曲线位于P“”中n次非奇异有理曲线上的锥面上，并且它们到P的投影提供了具有相同特征的曲线的例子，如果其次数与亏格相比足够小的话。例如，当n=4且C是属于关于P中以其顶点为中心的三次扭曲曲线上锥体吹起的线性系统mh-r(h\a超平面截面，r：一条直线)的非奇异不可约曲线时，从一般点将C(C在吹起下的同构像)投影到P所得到的曲线X对应于Hilb(P)的两个非约化不可约分支对于m^O的交点(见[4；定理3.1]和[5；命题B.2])。基本顺序(参看[2；定义1.4])
The open subscheme Hdig of Hilb(P ) which consists of points corresponding to nonsingular irreducible curves of degree d and genus g has more than one irreducible components in many cases. If one proceeds further to care about its connected components, he will necessarily encounter the problem whether red(Hdig.) is irreducible at the point corresponding to a given curve or not But even the examples of such reducible singular points of red(H^) do not seem to be known well, except that J. Harris in [6; p. 93] mentioned the existence of nondegenerate nonsingular irreducible curves in P" (n^4) whose Hilbert points lie on more than one irreducible components of Hilb(P"). These curves are on the cone over a nonsingular rational curve of degree n-l in P"" and the projection of them to P provides the examples of curves in P which have the same character, if the degree is sufficiently small as compared with the genus. For instance, when n = 4 and C is a nonsingular irreducible curve belonging to the linear system \mh-\-r (h\ a hyperplane section, r: a line of ruling) on the blowing up of the cone over a twisted cubic curve in P with center its vertex, the curve X obtained by projecting C (the isomorphic image of C under the blowing up) to P from a general point corresponds to a point of the intersection of two nonreduced irreducible components of Hilb(P) for m^O (see [4; Theorem 3. 1] and [5; Proposition B. 2]). The basic sequence (cf. [2; Definition 1.4])