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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].

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14C20 Divisors, linear systems, invertible sheaves