Following Faber–Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $\mathbb {P}^{1}$ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials.

We determine the integral Chow and cohomology rings of the moduli stack $\mathcal {B}_{r,d}$ of rank r, degree d vector bundles on $\mathbb {P}^1$ -bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ is a free $\mathbb {Q}$ -algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring $A^*(\mathcal {B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal {B}_{r,d})$ as a subring of $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ . From this description, we see that $A^*(\mathcal {B}_{r,d})$ is not finitely generated as a $\mathbb {Z}$ -algebra. Finally, when $k = \mathbb {C}$ , the cohomology ring of $\mathcal {B}_{r,d}$ is isomorphic to its Chow ring.

We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on a relation between tropical and complex intersection theories, which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.

We analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian–Oprea–Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian–Oprea–Pandharipande: that enumerative invariants defined as intersection numbers in the stable quotient moduli space coincide with Gromov–Witten invariants.

This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan in $N$, the choice consists of giving an inner product or a complete flag for ${{M}_{\mathbb{Q}}}\,=\,\mathbb{Q}\,\otimes \,\text{Hom(}N,\,\mathbb{Z}\text{)}$ , or more generally giving for each cone σ in the fan a linear subspace of ${{M}_{\mathbb{Q}}}$ complementary to ${{\sigma }^{\bot }}$ , satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. If $X$ is simplicial (for instance, if $X$ is non-singular), we obtain a commutative ring structure to the invariant cycles of $X$ with rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discuss how to define intersection cycles that require no choices, at the expense of increasing the size of the coefficient field.