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>）为特征为零的三维射影空间上积分曲线的饱和齐次理想的基本序列。(1)假设一个> = 2,b > = 1。设a‘和t为整数，且t+1<=a’<=a。如果n_i = n_t +它为所有我(t < = < = ')和n_ {+ 1} < n_{},然后t (t - 1) / 2 >马克斯{我| n_{+我}< n_{}, 1 < =我< = b}。(2)如果有一个整数(1 < =“< =),这样张n_i = n_1 +所有我(1 < = < =“),然后n_a”< = n_{+ 1}。(3)如果有一个整数(1 < =“< =),这样n_i = n_1 +我2为所有我(2 < = < =“),然后n_a”< = n_{+ 1}。以上三个不等式对包含在a次不可约化简曲面上的所有曲线都成立。我们还得到了下面的结果，在这里略去。(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在附加假设下对库克连通性猜想的证明是不完整的。我们对这个猜想给出了一个新的证明，在一个特殊的情况下包含了Decker-Schreyer的假设。序列a，n_1，…，n_{a+b}满足上述所有条件和其他已知条件，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.