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Ribbons and their canonical embeddings. (English) Zbl 0853.14016

What is the limit of the canonical model of a smooth curve as the curve degenerates to a hyperelliptic curve? “A ribbon” – more precisely “a ribbon on \(\mathbb{P}^1\)” – may be defined as the answer to this riddle. A ribbon on \(\mathbb{P}^1\) is a double structure on the projective line. Such ribbons represent a little-studied degeneration of smooth curves that shows promise especially for dealing with questions about the Clifford indices of curves.
The theory of ribbons is in some respects remarkably close to that of smooth curves, but ribbons are much easier to construct and work with. In this paper we discuss the classification of ribbons and their maps. In particular, we construct the “holomorphic differentials” – sections of the canonical bundle – of a ribbon, and study properties of the canonical embedding. Aside from the genus, the main invariant of a ribbon is a number we call the “Clifford index”, although the definition for it that we give is completely different from the definition for smooth curves. This name is partially justified here, and but much more so by two subsequent works: In the paper of L.-Y. Fong [J. Algebr. Geom. 2, No. 2, 295-307 (1995; Zbl 0788.14027)] a strong smoothing result for ribbons is proved. In the paper of D. Eisenbud and M. Green [Trans. Am. Math. Soc. 347, No. 3, 757-765 (1995; see the preceding review)] it is shown that the Clifford index of a ribbon may be re-expressed in terms of a certain notion of generalized linear series, and the semicontinuity of the Clifford index as a smooth curve degenerates to a ribbon is established. Together, these results imply that any ribbon may be deformed to a smooth curve of the same Clifford index.
Our original motivation for studying ribbons came from an attack on a conjecture of Mark Green concerning the free resolution of a canonical curve: If \(I \subset S = k[x_0, \dots, x_{g - 1}]\) is the homogeneous ideal of a canonically embedded curve \(C\) of arithmetic genus \(g\), then the free resolution of \(S/I\) is known to have the form \[ \begin{split} 0 \to S (- g - 1) \to S^{a_{g - 3}} (- g + 2) \oplus S^{b_{g - 3}} (- g + 1) \\ \to \cdots \to S^{a_i} (- i - 1) \oplus S^{b_i} (- i - 2) \to \cdots \to S^{a_1} (-2) \oplus S^{b_1} (-3) \to S \to S/I \to 0 \end{split} \] with \(a_{g - 2 - i} = b_i\) for all \(i\). The free modules notated \(S^{a_i} (- i - 1)\) above form a subcomplex. We define the resolution Clifford index of \(C\) to be the largest \(i\) for which \(a_i \neq 0\). – By contrast, the usual Clifford index of a smooth curve \(C\) of genus \(g \geq 3\) is defined as the maximum, over all line bundles \(L\) on \(C\) such that \(h^0(L)>1\) and \(h^1(L)>1\) of the quantity \[ \text{Cliff} L\;{\overset{def}=}\text{ degree } L_2 (h^0 (L) - 1) = g + 1 - h^0 (L) - h^1 (L). \] With this terminology, Green’s conjecture on canonical curves is the assertion that the Clifford index and the resolution Clifford index agree for smooth curves over an algebraically closed field of characteristic 0. In terms of the new Clifford index we define for a ribbon, we make the
Canonical ribbon conjecture: The resolution Clifford index and the Clifford index agree for ribbons over a field of characteristic 0.
Because of the smoothing results of Fong (loc. cit.) and Eisenbud-Green (loc. cit.), a proof of our conjecture for some ribbon of each genus and Clifford index would imply Green’s conjecture for a generic curve of each Clifford index. The restriction to characteristic 0 is really necessary in both cases, since examples of Schreyer for smooth curves and examples given below for ribbons show that the conjectures sometimes fail in finite characteristic.

MSC:

14H45 Special algebraic curves and curves of low genus
14C20 Divisors, linear systems, invertible sheaves
14Q05 Computational aspects of algebraic curves

Software:

Macaulay2
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References:

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