Related papers: Geometric Manin's conjecture in characteristic $p$
The Losev-Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and…
We derive conjectures, called genus 1 Enumerative mirror symmetry for moduli spaces of Higgs bundles, which relate curve-counting invariants of moduli spaces of Higgs $\mathrm{SL}_r$-bundles to curve-counting invariants of moduli spaces of…
We propose an analogue of Dubrovin's conjecture for the case where Fano manifolds have quantum connections of exponential type. It includes the case where the quantum cohomology rings are not necessarily semisimple. The conjecture is…
We construct the Gromov-Witten invariants of moduli of stable morphisms to $\Pf$ with fields. This is the all genus mathematical theory of the Guffin-Sharpe-Witten model, and is a modified twisted Gromov-Witten invariants of $\Pf$. These…
We describe a very large class of conjectural relations in the tautological ring of the moduli space $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, extending and generalizing the Faber-Zagier relations. These notes…
Manin's Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin's Conjecture is a thin set.
We define a new height function on rational points of a DM (Deligne-Mumford) stack over a number field. This generalizes a generalized discriminant of Ellenberg-Venkatesh, the height function recently introduced by…
We study the fix point components of the big torus action on the moduli space of stable maps into a smooth projective toric variety, and apply Graber and Pandharipande's localization formula for the virtual fundamental class to obtain an…
We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that…
Inspired by a method of La Bret\`eche relying on some unique factorisation, we generalize work of Blomer, Br\"udern, and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of…
We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted…
The purpose of this paper is to formulate a number of conjectures giving a rather complete description of the tautological ring of M_g and to discuss the evidence for these conjectures.
We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a…
We formulate a general conjecture relating Chern classes of subbundles of Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin L-functions of number fields. This conjecture may be viewed as a far-reaching…
We show that moduli spaces of transversely cut-out (perturbed) pseudo-holomorphic curves in an almost complex manifold carry canonical relative smooth structures ("relative to the moduli space of domain curves"). The main point is that…
In this paper we extend the discussion on Homological Mirror Symmetry for Fano toric varieties presented by Hori and Vafa to more general case of monotone symplectic manifolds with real polarizations. We claim that the Hori -- Vafa…
In their paper "Quantum cohomology of projective bundles over $P^n$" (Trans. Am. Math. Soc. (1998)350:9 3615-3638) Z.Qin and Y.Ruan introduce interesting techniques for the computation of the quantum ring of manifolds which are…
A conjectural recursive relation for the Poincar\'e polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also…
On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or…
This is a survey paper on Geometric Manin's conjecture which was proposed by Brian Lehmann and the author. We introduce Geometric Manin's conjecture (GMC) and review some recent progress on this conjecture.