Monomial space curves in ³ as set-theoretic complete intersections
Monomial space curves in ³ as set-theoretic complete intersections
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³ 中的单项式空间曲线作为集合论完全交集
DOI：
10.1090/s0002-9939-1979-0529205-5
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
H. Bresinsky
中科院分区：
文献类型：
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作者：
H. Bresinsky
构造性地证明了3-仿射空间中的单式空间曲线都是集合论完全交。J. Herzog（私人通信）证明了任意域K上仿射3-空间A3中的所有空间曲线，参数给定为x1 = t '，x2 = tf，X3 = t0，a，8，-y正整数，g.c.d.（a，，B，-y）= 1，是完备的集合论交。以下独立证明提供了一种算法来显式确定所涉及的表面，并且仅使用“高中代数方法”。“很明显，我们只需要考虑指定类型的空间曲线，它们不是理想理论上的完全相交。- 通过[1]，定义这样的曲线C的素理想P C K[x1，X2，X3]由p =（fi = X1 2 3 2 1 3 3 1 2）给出，其中所有指数都是整数，大于0并且满足关系a1 = a21 + a31，a2 = a12 + a32，a3 = a13 + a23。我们称D =（f1，f2，f3）n（X ~（12），X ~（22）f3，X ~（22）f3）=（f1，f2）.证据第一个平等：2是明确的。如果I = ggj E D，则g3 f3 E（xa 21，xa 12）。根据[2]，（X '21，Xa 12）是不可约的，因此是初等的。由于f34（x1，X2）=（2，X2），al）g3 E（XC 121，XC 112），由此C .第二个等式：2是平凡的。简单计算表明：x2 f = -xa 37 f1 XaI 2f 2X212 f3 = X13 f2 X3 a23 f1，由此可得C .因此，C和具有等式xl = 0、x2 = 0的线1是（f1，f2）的零点。由于Cn i = {（0，0，0）），构造多项式g E P使得ft e（g fi），g = X3 + h，h E（x1，x2）就足够了.为了实现这一点，我们采取（x2 -xl 21 X223）一个X22 k？x 3x 23，减去或加上x I（a22 O）X 1a 23 f，然后除以X“a12。这就给出了x2 a-C2 k？X1（a2 - 1）XC'13+ a1 a23 EP.如果a21 = 1，我们就完成了;如果不是，我们表明，经过适当的修改，这个过程可以进行a21次。为此，考虑术语X12 a2 X（a1 -i）a21 X（a1 i）G23，1？j a211，从f2 '的二项式展开获得。由于（a21，j）a1 <a21，a1-ja 21，因此可以通过将fi的适当倍数减去xJa 2+（a2，1）a2 Xa 3-j）a212-（a2，-j）ja 1，1-j）a23+（a2，-J）a3来改变该项。AMS（MOS）主题分类表-v（1970年）。小学14 M10;中学13 A15。
It is shown constructively that all monomial space curves in affine 3-space are set-theoretic complete intersections. It was shown by J. Herzog (private communication) that all space curves in affine 3-space A3 over an arbitrary field K, given parametrically by xl = t', x2 = tf, X3 = t0, a, 8, -y positive integers, g.c.d.(a, ,B, -y) = 1, are complete set-theoretic intersections. The following independent proof provides an algorithm to determine the surfaces involved explicitly and uses only "high-school algebra methods." It is clear that we only have to consider space curves of the indicated type, which are not ideal-theoretic complete intersections. -By [1], the prime ideal P C K[x1, X2, X3], defining such a curve C is given by p = (fi = X1 2 3 2 2 1 3 3 3 1 2 where all exponents are integral, greater than 0 and satisfy the relations a1 = a21 + a31, a2 = a12 + a32, a3 = a13 + a23. We claim: D = (f1,f2,f3) n (X12, X2'2) = (f3,f2, x 22f3, X2f3) = (fl,f2). PROOF. 1st equality: 2 is clear. Iff I = ggj E D, then g3f3 E (X a2l, xa 12). By [2], (X'21, Xa12) is irreducible, hence primary. Since f3 4 (xl, X2) = (2 , X~2) ,al) g3 E (XC121, XC112), from which C . 2nd equality: 2 is trivial. An easy calculation shows x2f = -x a37f1 Xa I2f2 X212f3 = X13f2 X3a23f1, from which C . Therefore C and the line 1 with equations xl = 0, x2 = 0 are the zeroes of (fi, f2). Since C n / = {(O, 0, 0)), it suffices to construct a polynomial g E P such thatft e (g fi), g = X3 + h, h E (x1, x2). To accomplish this we take (x2 -xl21X223)a X22k ? x 3x23, subtract or add x I(a22 O)X 1a23f, and divide by X"a12. This gives x 2a -C2k ? X1(a2 -l)XC'13+a1a23 E P. If a21 = 1 we are done; if not we show that the process, after proper modification, can be carried through a2l-times. To this end, consider the term Xi2a2X(a1 -i)a21X(a1 i)G23, 1 ? j a21 1, obtained from the binomial expansion of f2'. Since (a21 j)al < a21a1 -ja2l, this term can be changed by subtracting proper multiples of fi into xJa2+(a2 I)a 2Xa 3-j)a212-(a2, -])ja, l-j)a23+ (a2,-J)a3 Received by the editors Ju-ne 10, 1978. AMS (MOS) subject classifica.. -v (1970). Primary 14M10; Secondary 13A15.