## Curves of minimal genus on a general abelian variety.(English)Zbl 0864.14027

Let $$C$$ be a smooth projective curve and let $$\varphi:C\to A$$ be a morphism, with $$A$$ abelian variety. If the image $$\varphi(C)$$ generates $$A$$, then $$A$$ is isomorphic to a quotient of the Jacobian of $$C$$; in general, one finds infinitely many curves $$C$$ which map to $$A$$ as above and the minimal genus of these curves is called the jacobian dimension of $$C$$. Several lower bounds are known for the jacobian dimension of a general abelian variety $$A$$. The authors are mainly concerned here with the case $$\dim(A)=4,5$$; they show that a general abelian variety of dimension 5 has jacobian dimension 11; for general abelian fourfolds $$A$$, they prove that the jacobian dimension is 7, furthermore they characterize curves of genus 7 mapping to $$A$$ as above and compute their number.

### MSC:

 14K30 Picard schemes, higher Jacobians 14H45 Special algebraic curves and curves of low genus

### Keywords:

abelian variety; jacobian dimension
Full Text:

### References:

 [1] E. Arbarello , M. Comalba , P. Griffiths and J. Harris , Geometry of algebraic curves. I , Grundlehren der Math. 267, Springer-Verlag, 1984. · Zbl 0559.14017 [2] A. Alzati and P. Pirola , On abelian subvarieties generated by symmetric correspondences , Math. Z. 205 (1990), 333-342. · Zbl 0685.14018 [3] F. Bardelli , Curves of genus three on a general abelian threefold and the non finite generation of the Griffiths group, in Arithmetic of Complex Manifolds , Proceedings, Erlangen, 1988, ed. W. Barth, H. Lange, Springer Lecture Notes in Math., 1399 (1989), 10-26. · Zbl 0703.14004 [4] A. Beauville , Prym varieties and the Schottky problem , Invent. Math. 41 (1977), 149-196. · Zbl 0333.14013 [5] Ch. Birkenhake and H. Lange , Complex abelian varieties , Grundlehren der math. Wiss., Springer-Verlag, 1992. · Zbl 0779.14012 [6] C. Ciliberto , Alcune applicazioni di un classico procedimento di Castelnuovo , Sem. di variabili Complesse, Univ. di Bologna, 1982-83,17-43. · Zbl 0612.14028 [7] L. Chiantini and C. Ciliberto , A few remarks on the lifting problem , preprint (1992). · Zbl 0813.14043 [8] C. Ciliberto , G. Van Der Geer and M. Teixidor , On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms , J. Alg. Geom. 1 (1992), 215-229. · Zbl 0806.14020 [9] C. Ciliberto and G. Van Der Geer , Subvarieties of the moduli space of curves parametrizing Jacobians with non-trivial endomorphisms , Am. J. Math. 114 (1991), 551-570. · Zbl 0766.14016 [10] H. Clemens , Double solids , Adv. in Math. 47 (1983). · Zbl 0509.14045 [11] R. Donagi , The tetragonal construction , Bull. Amer. Math. Soc. 4 (1981), 181-185. · Zbl 0491.14016 [12] O. Debarre , Degree of curves in abelian varieties , preprint (1992). · Zbl 0781.14031 [13] R. Donagi and R. Smith , The structure of the Prym map , Acta Math. 146 (1981), 25-102. · Zbl 0538.14019 [14] Ph. Griffiths , Periods of integrals over algebraic manifolds, III , Publ. Math. IHES, 38 (1970), 125-180. · Zbl 0212.53503 [15] V. Kanev , Theta divisors of generalized Prym varieties I , Springer Lecture Notes in Math. 1124 (1985), 166-215. · Zbl 0575.14037 [16] V. Kanev , Principal polarizations of Prym-Tjurin varieties , preprint ( ). · Zbl 0694.14009 [17] D. Mumford , Prym varieties I, Contribution to Analysis , Academic Press, 1974. · Zbl 0299.14018 [18] G.P. Pirola , Abel-Jacobi invariant and curves on generic abelian varieties , preprint (1992). · Zbl 0837.14036 [19] S. Recillas , Jacobian of curves with a g1 4 are Prym varieties of trigonal curves , Bol. Soc. Mat. Mexicana, 19 (1974), 9-13. · Zbl 0343.14012 [20] F. Severi , Sulle corrispondenze fra i punti di una curva algebrica e sopra certe classi di superficie , Mem. Accad. Sci.Torino, (2) 54 (1903). · JFM 35.0629.01 [21] E. Sernesi , Neutral linear series , preprint, Erlangen (1988). [22] R. Torelli , Sulle serie algebriche semplicemente infinite di gruppi di punti appartenenti a una curva algebrica , Rend. Circ. Mat. Palermo, 37 (1914). · JFM 45.0816.02 [23] G. Welters , Curves of twice the minimal class on principally polarized abelian varieties , Indagationes Math. 49 (1987), 87-109. · Zbl 0644.14014
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