## Duke Mathematical Journal

### An interesting 0-cycle

#### Abstract

The geometric and arithmetic properties of a smooth algebraic variety $X$ are reflected by the configuration of its subvarieties. A principal invariant of these are the Chow groups $\CH^p(X)$, defined to be the group of codimension-$p$ algebraic cycles modulo rational equivalence. For $p=1$ these groups are classical and well understood. For $p\dgeqq 2$ they are nonclassical in character and constitute a major area of study. In particular, it is generally difficult to decide whether a given higher codimension cycle is or is not rationally equivalent to zero. In their study of the moduli spaces of algebraic curves, C. Faber and R. Pandharipande introduced a canonical $0$-cycle $z_K$ on the product $X=Y\times Y$ of a curve $Y$ with itself. This cycle is of degree zero and Albanese equivalent to zero, and they asked whether or not it is rationally equivalent to zero. This is trivially the case when the genus $g=0,1,2$, and they proved that this is true when $g=3$. It is also the case when $Y$ is hyperelliptic or, conjecturally, when it is defined over a number field. We show that $z_K$ is not rationally equivalent when $Y$ is general and $g\dgeqq 4$. The proof is variational, and for it we introduce a new computational method using Shiffer variations. The condition $g\dgeqq 4$ enters via the property that the tangent lines to the canonical curve at two general points must intersect.

#### Article information

Source
Duke Math. J., Volume 119, Number 2 (2003), 261-313.

Dates
First available in Project Euclid: 23 April 2004

https://projecteuclid.org/euclid.dmj/1082744733

Digital Object Identifier
doi:10.1215/S0012-7094-03-11923-7

Mathematical Reviews number (MathSciNet)
MR1997947

Zentralblatt MATH identifier
1058.14014

#### Citation

Green, Mark; Griffiths, Phillip. An interesting 0-cycle. Duke Math. J. 119 (2003), no. 2, 261--313. doi:10.1215/S0012-7094-03-11923-7. https://projecteuclid.org/euclid.dmj/1082744733

#### References

• A. Ching, On the generalized Noether-Lefschetz locus, preprint.
• H. Esnault and K. H. Paranjape, Remarks on absolute de Rham and absolute Hodge cycles, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 67--72.
• M. L. Green, Griffiths' infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom. 29 (1989), 545--555.
• M. Green and P. Griffiths, Hodge-theoretic invariants for algebraic cycles, Internat. Math. Res. Notices 2003, no. 9, 477--510. \CMP1 951 543
• M. Green, J. Murre, and C. Voisin, Algebraic Cycles and Hodge Theory (Torino, Italy, 1993), Lecture Notes in Math. 1594, Springer, Berlin, 1994.
• D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195--204.
• M. V. Nori, Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), 349--373.
• M. Saito, Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), 283--331.
• S. Saito, Motives, algebraic cycles and Hodge theory'' in The Arithmetic and Geometry of Algebraic Cycles (Banff, Alberta, 1998), CRM Proc. Lecture Notes 24, Amer. Math. Soc., Providence, 2000, 235--253.
• V. Srinivas, Gysin maps and cycle classes for Hodge cohomology, Proc. Indian Acad. Sci. Math. Sci. 103 (1993), 209--247.
• C. Voisin, Transcendental methods in the study of algebraic cycles'' in Algebraic Cycles and Hodge Theory (Torino, Italy, 1993), Lecture Notes in Math. 1594, Springer, Berlin, 1994, 153--222.
• --. --. --. --., Variations de structure de Hodge et zéro-cycles sur les surfaces générales, Math. Ann. 299 (1994), 77--103.
• --. --. --. --., Some results on Green's higher Abel-Jacobi map, Ann. of Math. (2) 149 (1999), 451--473.