Study of basic sequences of integral curves in three dimensional projective spaces

设(a；n_1，n_2，…，n_a；n_{a+1}，...，n_(a+b))(b&gt；0)是特征为零的无限域上三维射影空间中积分曲线的饱和齐次理想的基本序列.(1)设a&gt；=2，b&gt；=1.设a‘和t是整数，t+1&lt；=a’&lt；=a，若n_i=n_t+i-t，对所有i(t&lt；=i&lt；=a‘)和n_{a+1}&lt；n_{a’}，则t(t-1)/2&gt；max{i|n_{a+i}&lt；n_{a‘}，1&lt；=i&lt；=b}.(2)If存在整数a’(1&lt；=a‘&lt；=a)，使得对所有i(1&lt；=i&lt；=a’)，n_i=n_1+i-1，则n_a‘&lt；=n_{a+1}。(3)如果存在一个整数a‘(1&lt；=a’&lt；=a)使得对于所有i(2&lt；=i&lt；=a‘)，n_i=n_1+i-2，则n_a’&lt；=n_(a+1).上述三个不等式对包含在a次不可约约化曲面中的所有曲线都是有效的.我们还得到了下面的结果，这里省略它的细节.(4)b=1的积分曲线的基本序列必须满足的必要条件.(5)Decker-Schreyer在附加假设下对Cook连通性猜想的证明是不完全的.对于包含Decker-Schreyer假设的一种特殊情况，我们给出了该猜想的一个新的证明，满足上述所有条件的序列a，n_1，…，n_(a+b)都可以通过我们的计算机程序得到。

Let (a;n_1,n_2,...,n_a;n_{a+1},...,n_{a+b})(b>0) be the basic sequence of the saturated homogeneous ideal of an integral curve in a three dimensional projective space over an infinite field of characteristic zero.(1)Suppose a>=2,b>=1. Let a' and t be integers with t+1<=a'<=a. If n_i=n_t+i-t for all i(t<=i<=a') and n_{a+1}<n_{a'}, then t(t-1)/2>max{i|n_{a+i}<n_{a'},1<=i<=b}.(2)If there is an integer a'(1<=a'<=a) such that n_i=n_1+i-1 for all i(1<=i<=a'), then n_a'<=n_{a+1}.(3)If there is an integer a'(1<=a'<=a) such that n_i=n_1+i-2 for all i(2<=i<=a'), then n_a'<=n_{a+1}.The above three inequalities are valid for all curves which is contained in an irreducible reduced surface of degree a. We have also obtained the results below, whose details are omitted here.(4)Necessary conditions that the basic sequence of an integral curve with b=1 must satisfy.(5)The proof given by Decker-Schreyer of the Cook's connectedness conjecture under additional assumptions turned outs to be incomplete. We gave a new proof to that conjectur for a special case embracing the case set up by Decker-Schreyer's assumptions.The sequences a,n_1,...,n_{a+b} satisfying all the above conditions and others known so far can be obtained by our computer program.