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Let C be a complex curve of genus g, let J(C) be its Jacobian and let R(C) be its tautological ring, that is, the group of algebraic cycles modulo algebraic equivalence. We study the algebraic structure of R(C). In particular, we give a detailed description of all the possibilities that may occur for g<9: we construct convenient basis and we determine the matrices representing the Fourier transform and both intersection and Pontryagin products explicitly. In particular, we estimate the dimension of R(C).
Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying the following condition: the line bundle f*(S)
We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $leq 6$. In particula
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety $X$ as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into $X$ by polynomial reparametrisations. Using
In this paper, we discuss the cycle theory on moduli spaces $cF_h$ of $h$-polarized hyperkahler manifolds. Firstly, we construct the tautological ring on $cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on m
Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $ell$-adic tautological systems for the case where the ground field is of characteristic $p$.