# The Chow rings of moduli spaces of elliptic surfaces over $\mathbb{P}^1$

@inproceedings{Canning2021TheCR, title={The Chow rings of moduli spaces of elliptic surfaces over \$\mathbb\{P\}^1\$}, author={Samir Canning and Bochao Kong}, year={2021} }

Let EN denote the coarse moduli space of smooth elliptic surfaces over P 1 with fundamental invariant N . We compute the Chow ring A(EN ) for N ≥ 2. For each N ≥ 2, A(EN ) is Gorenstein with socle in codimension 16, which is surprising in light of the fact that the dimension of EN is 10N − 2. As an application, we show that the maximal dimension of a complete subvariety of EN is 16. When N = 2, the corresponding elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice U . We show…

## References

SHOWING 1-10 OF 23 REFERENCES

Quadric ramk loci on moduli of curves and K3 surfaces

- Mathematics
- 2017

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed…

The Moduli of Weierstrass Fibrations Over IP 1

- 1981

Let k be an algebraically closed field of characteristic 4 = 2, 3. Let X p , Y be a flat proper map of reduced irreducible k-schemes such that every geometric fibre is either (a) an elliptic curve,…

Note on tautological classes of moduli of K3 surfaces

- Mathematics
- 2005

In this note we prove some cycle class relations on moduli of K3 surfaces. This note deals with a few properties of tautological classes on moduli spaces of K3 surfaces. Let M2d denote a moduli stack…

The Chow ring of a classifying space

- Mathematics
- 1998

We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to…

Segre classes and Hilbert schemes of points

- Mathematics
- 2015

We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via…

Relations in the tautological ring of the moduli space of $K3$ surfaces

- MathematicsJournal of the European Mathematical Society
- 2019

We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler's relation in genus 1, we construct a universal…

Tautological classes on moduli spaces of hyper-Kähler manifolds

- MathematicsDuke Mathematical Journal
- 2019

We study algebraic cycles on moduli spaces F h of h-polarized hyperkahler manifolds. Following previous work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3…

Gromov-Witten theory and Noether-Lefschetz theory

- Mathematics
- 2007

Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3…

The Noether-Lefschetz conjecture and generalizations

- Mathematics
- 2014

We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes…

3264 and All That: A Second Course in Algebraic Geometry

- Mathematics
- 2016

Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. Singular…