多个原始骨丘与骨丘变形
Courbes multiples primitives et déformations de courbes lisses
摘要：
本原多重曲线是复数域上的Cohen-Macaulay概型Y，使得约化概形C=Y_red是平滑曲线，并且Y可以局部嵌入光滑曲面。一般来说，这样的曲线Y不能全局嵌入光滑表面中。若Y是重数为n的本原重曲线，则存在Y的典范滤子C=C_1. C_n=Y使得C_i是重数为i的本原重曲线. Y中C的理想层I_C是C_{n-1}上的线丛。设T是一条平滑曲线，t_0是T的闭点。设D-->T是一族平坦的射影光滑不可约曲线，C= D_{t_0} .则C在D中的第n个无穷小邻域是嵌在光滑曲面D中的重数为n的本原重曲线C_n，此时I_C是C_{n-1}上的平凡线丛.反之，我们证明了C_{n-1}上每一条使I_C为平凡线丛的射影本原重曲线Y=C_n都可以用这种方法得到。
英文摘要：
A primitive multiple curve is a Cohen-Macaulay scheme Y over the field of complex numbers such that the reduced scheme C=Y_red is a smooth curve, and that Y can be locally embedded in a smooth surface. In general such a curve Y cannot be globally embedded in a smooth surface. If Y is a primitive multiple curve of multiplicity n, then there is a canonical filtration of Y C=C_1 ... C_n=Y such that C_i is a primitive multiple curve of multiplicity i. The ideal sheaf I_C of C in Y is a line bundle on C_{n-1}. Let T be a smooth curve and t_0 a closed point of T. Let D-->T be a flat family of projective smooth irreducible curves, and C=D_{t_0}. Then the n-th infinitesimal neighbourhood of C in D is a primitive multiple curve C_n of multiplicity n, embedded in the smooth surface D, and in this case I_C is the trivial line bundle on C_{n-1}. Conversely, we prove that every projective primitive multiple curve Y=C_n such that I_C is the trivial line bundle on C_{n-1} can be obtained in this way.
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