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The Clifford dimension of a projective curve. (English) Zbl 0703.14020

Let C be a curve (always projective and smooth) and L a line bundle on C. Then \(text{Cliff}(L):=\deg (L)-2h^ 0L+2.\) This is a measure of how unusual the line bundle is. We define Cliff(C) as the minimum, taken over all L, such that \(h^ 0L\) and \(h^ 1L\geq 2,\) of Cliff(L). We can now define the Clifford dimension of C as \(r=\min \{h^ 0L-1| \quad text{Cliff}(L)=text{Cliff}(C)\}.\) It is easy to prove that if \(r>1\), any line bundle L such that \(text{Cliff}(L)=text{Cliff}(C)\) is very ample. Thus if \(r>1\), a curve of Clifford dimension r can be seen as a curve embedded in \({\mathbb{P}}^ r\). We call such an embedding a Clifford embedding and the corresponding line bundle a Clifford bundle.
The case \(r=1\) and \(r=2\) are well-known: a curve C (with \(text{Cliff}(C)=c)\) is of Clifford dimension 1 if and only if C is \((c+2)\)-gonal; a curve C is of Clifford dimension 2 if and only if C is a plane curve of degree \(\geq 5.\)
We now assume that \(r\geq 3\). It is believed that Clifford curves in \({\mathbb{P}}^ r\) are extremely rare. For example G. Martens has proved that for \(r=3\) we only have the complete intersection of two cubics in \({\mathbb{P}}^ 3\). - So the first question is to construct Clifford curves for every r. The authors prove a recognition theorem; if C is a linearly normal curve of genus \(g=4r-2\) and of degree \(d=g-1\) in \({\mathbb{P}}^ r\) then the following are equivalent: C is a Clifford curve; C is not contained in any quadric of rank \(\leq 4;\) C is 2r-gonal. Further in this case, C is half-canonical. The authors construct, for all r, such curves by using some linear systems on some special K3 surfaces (these surfaces are obtained by using the surjectivity of the period map).
The authors conjecture that all the Clifford curves of Clifford dimension \(r\geq 3\) have genus \(4r-2,\) Clifford index \(2r-3\) and degree \(4r-3.\) This conjecture is related to the Castelnuovo formula which gives the number \(C(d,g,r)\) of \((2r-2)\)-secant \((r-2)\)-planes of a projective curve of degree d and genus g: if there exists a Clifford curve in \({\mathbb{P}}^ r\) with genus g and degree d, then \(C(d,g,r)=0\). Before resolving these equations, the authors find a bound on the genus and the degree (depending on r) of any Clifford curve in \({\mathbb{P}}^ r\). Now they want to resolve \(C(d,g,r)=0\) in this range and eliminate those g and d which do not correspond to smooth projective curves in \({\mathbb{P}}^ r\). But the Castelnuovo formula is very complicated and the computational part of this program is completely done only for \(3\leq r\leq 9\). So in this case the principal conjecture (explained above) is proved.
Reviewer: J.-Y.Merindol

MSC:

14H45 Special algebraic curves and curves of low genus
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H10 Families, moduli of curves (algebraic)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N05 Projective techniques in algebraic geometry
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