Order sequences and rational curves.

*(English)*Zbl 0846.14023
Ballico, Edoardo (ed.), Projective geometry with applications. A collection of 15 research papers. New York, NY: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 166, 27-42 (1994).

The theory of Weierstrass points on a smooth curve in arbitrary characteristic was initiated by F. K. Schmidt about half a century ago. In the positive characteristic theory of Weierstrass points, the order-sequence plays an important role as an invariant of a linear system. A fundamental and combinatorial property of order-sequences in characteristic \(p > 0\) is to satisfy the \(p\)-adic criterion, which is stated as follows. Let \(b_0 < b_1 < \cdots < b_N \) be a sequence of nonnegative integers.

\(p\)-adic criterion. If a nonnegative integer \(m\) satisfies \({b_j \choose m} \not \equiv 0 \bmod p\) for some \(j\), then \(m = b_i\) for some \(i\).

Conversely, any sequence of nonnegative integers \(b_0 < b_1 < \cdots < b_N\) satisfying the \(p \)-adic criterion is the order-sequence of a linear system on a curve. In fact, the sequence is the order-sequence of the linear system on \(\mathbb{P}^1\) corresponding to the morphism \(\varphi = \varphi^{\langle b_0 \dots b_N \rangle} : \mathbb{P}^1 \to \mathbb{P}^N\) defined by \(\varphi (t) = (t^{b_0}, \dots, t^{b_N})\). – In this paper, we direct our attention to nondegenerate rational curves in projective spaces, or linear systems on \(\mathbb{P}^1\) and study them in a context of order-sequences.

Theorem 1: Let \({\mathcal G}\) be a base-point-free linear system of degree \(d\) and of dimension \(N > 0\) on a curve of genus \(g\) over an algebraically closed field of characteristic 0. Denote by \(\nu\) the (set-theoretic) number of \({\mathcal G}\)-Weierstrass points. Then \((\nu - 2) ({d \over N} - 1) \geq 2g\). When \({\mathcal G}\) is the canonical linear system, the inequality means \(\nu \geq 2g + 2\), which is sharp. When \({\mathcal G} = g^1_d\), it says \(\nu \geq {2 \over d - 1} g + 2\), which is almost sharp by Hurwitz formula. In section 4, we establish a characteristic-free version of theorem 1:

If a base-point-free linear system \(g_d^N\) with order-sequence \(b_0, \dots, b_N\) is tame and \(Z^{(N - 1)}\) for \(g^N_d\) is empty, then the inequality \((\nu - 2) ({d \over b_N } - 1) \geq 2g\) holds except for linear systems corresponding to \(\varphi^{\langle b_0 \dots b_N \rangle}\)’s.

For the entire collection see [Zbl 0801.00005].

\(p\)-adic criterion. If a nonnegative integer \(m\) satisfies \({b_j \choose m} \not \equiv 0 \bmod p\) for some \(j\), then \(m = b_i\) for some \(i\).

Conversely, any sequence of nonnegative integers \(b_0 < b_1 < \cdots < b_N\) satisfying the \(p \)-adic criterion is the order-sequence of a linear system on a curve. In fact, the sequence is the order-sequence of the linear system on \(\mathbb{P}^1\) corresponding to the morphism \(\varphi = \varphi^{\langle b_0 \dots b_N \rangle} : \mathbb{P}^1 \to \mathbb{P}^N\) defined by \(\varphi (t) = (t^{b_0}, \dots, t^{b_N})\). – In this paper, we direct our attention to nondegenerate rational curves in projective spaces, or linear systems on \(\mathbb{P}^1\) and study them in a context of order-sequences.

Theorem 1: Let \({\mathcal G}\) be a base-point-free linear system of degree \(d\) and of dimension \(N > 0\) on a curve of genus \(g\) over an algebraically closed field of characteristic 0. Denote by \(\nu\) the (set-theoretic) number of \({\mathcal G}\)-Weierstrass points. Then \((\nu - 2) ({d \over N} - 1) \geq 2g\). When \({\mathcal G}\) is the canonical linear system, the inequality means \(\nu \geq 2g + 2\), which is sharp. When \({\mathcal G} = g^1_d\), it says \(\nu \geq {2 \over d - 1} g + 2\), which is almost sharp by Hurwitz formula. In section 4, we establish a characteristic-free version of theorem 1:

If a base-point-free linear system \(g_d^N\) with order-sequence \(b_0, \dots, b_N\) is tame and \(Z^{(N - 1)}\) for \(g^N_d\) is empty, then the inequality \((\nu - 2) ({d \over b_N } - 1) \geq 2g\) holds except for linear systems corresponding to \(\varphi^{\langle b_0 \dots b_N \rangle}\)’s.

For the entire collection see [Zbl 0801.00005].

##### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14C20 | Divisors, linear systems, invertible sheaves |