## 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
Full Text:

### References:

 [1] R.D.M. Accola : Plane models for Riemann surfaces admitting certain half-canonical linear series, part I , in Riemann Surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, ed. I. Kra and B. Maskt, Annals of Math. Studies 97, pp 7-20, Princeton University Press 1981. · Zbl 0491.30038 [2] E. Arbarello , M. Cornalba , P.A. Griffith , J. Harris : Geometry of Algebraic curves Vol I, Grundl, d. math. Wiss. 267, Springer Verlag (1985). · Zbl 0559.14017 [3] D.A. Buchsbaum , D. Eisenbud : Algebra structures for finite free resolutions and some structure theorems for ideals in codimension 3 . Am. J. Math. 99(1977), 447-485. · Zbl 0373.13006 [4] W. Barth , C. Peters , A. Van De Ven : Compact complex surfaces , Erg. d. Math. u. Grenzg. 3 Folge 4, Springer Verlag (1984) · Zbl 0718.14023 [5] D. Bayer , M. Stillman : Macaulay, a computer algebra system, 1981-1989 . Available free for a variety of computers from D. Bayer, Dept. of Math. Columbia Univ. or M. Stillman, Dept. of Math., Cornell Univ. [6] E. Ballico : On the Clifford index of algebraic curves , Proc. Am. Math. Soc. 97 (1986), 217-218. · Zbl 0591.14020 [7] D. Castelnuovo : Una applicazione delle geometria ennumerativa alle curve algebraique, Rendic . Palermo 3^\circ .89 · JFM 21.0674.02 [8] D. Eisenbud , J. Harris : Curves of almost maximal genus . Les Presses de l’université de Montréal (1982), 81-138. [9] H.M. Farkas : Remarks on Clifford indices . J. Reine Angew. Math. 391 (1988), 213-220. · Zbl 0651.14018 [10] M. Green : Koszul cohomology of projective varieties I , J. Diff. Geometry 19 (1984), 127-171. · Zbl 0559.14008 [11] M. Green , R. Lazarsfeld : On the projective normality of complete linear series on an algebraic curve , Inv. Math. 83 (1986), 73-90. · Zbl 0594.14010 [12] M. Green , R. Lazarsfeld : Special divisors on K3-surfaces , Inv. Math. 89 (1987), 357-370. · Zbl 0625.14022 [13] T. Horowitz : Varieties of low \Delta -genus , Duke Math. J. 50 (1983), 667-683. · Zbl 0536.14021 [14] G. Martens : Funktionen von vorgegebener Ordnung auf komplexen Kurven , J. Reine Angew. Math. 320 (1980), 68-85. · Zbl 0441.14010 [15] G. Martens : Über den Clifford Index algebraischer Kurven , J. Reine Angew. Math. 336 (1982), 83-90. · Zbl 0484.14010 [16] G. Martens : On Dimension Theorems of the Varieties of Special Divisors on a Curve , Math. Ann. 267(1984), 279-288. · Zbl 0519.14021 [17] G. Martens : On curves on K3-surfaces . to appear. · Zbl 0698.14036 [18] A. Mayer : Families of K3-surfaces , Nayoga Math. J. 48 (1972), 107-111. · Zbl 0244.14012 [19] T. Meis : Die minimale Blätterzahl der Konkretisierungen einer kompakten Riemannschen Fläsche . Schriftenreihe des Math. Inst. d. Univ. Münster 16 (1960). · Zbl 0093.07603 [20] D.R. Morrison : On K3-surfaces with large Picard number . Inv. Math. 75 (1984),105-121. · Zbl 0509.14034 [21] C. Okonek : Moduli reflexiver Garben und Flächen von kleinem Grad in P4 , Math. Z. 184 (1983), 549-572. · Zbl 0524.14018 [22] J. Riordan : Combinatorial Identities . John Wiley & Sons (1968) · Zbl 0194.00502 [23] C.L. Siegel : Über einige Anwendungen diophantischer Approximationen , Abh. Preuss. Akad. Wiss.,Phys. -Math. KL., 1929, Nr. 1. · JFM 56.0180.05 [24] F.O. Schreyer : Syzygies of canonical curves and special linear series , Math. Ann. 275 (1986), 105-137. · Zbl 0578.14002 [25] B. Saint-Donat : Projective models of K3-surfaces , Amer. J. Math. 96 (1974), 602-639. · Zbl 0301.14011 [26] A. Sommese : Hyperplane sections of projective surfaces I The adjunction mapping , Duke Math. J. 46 (1979), 377-401. · Zbl 0415.14019 [27] G. Szegö : Orthogonal polynomials , Amer. Math. Soc. Coll. Publ. XXIII, New York (1959). · Zbl 0089.27501 [28] A. Van De Ven : On the 2-connectedness of very ample divisors on a surface , Duke Math. J. 46 (1979), 403-406. · Zbl 0458.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.