Gaussian rational points on a singular cubic surface

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manin's conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manin's conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.Comment: 16 page

Higher-genus wall-crossing in the gauged linear sigma model

We introduce a technique for proving all-genus wall-crossing formulas in the gauged linear sigma model as the stability parameter varies, without assuming factorization properties of the virtual class. We implement this technique explicitly for the hybrid model, which generalizes our previous work to the Landau--Ginzburg phase.Comment: 42 pages, 2 figures. v2: Added an appendix (by Yang Zhou) proving the n=0 case of the main theorem; additional substantial changes to Sections 2.5 (broad vanishing) and 4.3 (localization contributions

Powers of the theta divisor and relations in the tautological ring

We show that the vanishing of the $(g+1)$-st power of the theta divisor on the universal abelian variety $\mathcal{X}_g$ implies, by pulling back along a collection of Abel--Jacobi maps, the vanishing results in the tautological ring of $\mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber--Vakil, and Faber--Pandharipande. We also show that Pixton's double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem~$\star$ of Graber and Vakil, and we provide an explicit algorithm for expressing any tautological class on $\overline{\mathcal{M}}_{g,n}$ of sufficiently high codimension as a boundary class.Comment: 29 page

Towards a Theory of Logarithmic GLSM Moduli Spaces

In this article, we establish foundations for a logarithmic compactification of general GLSM moduli spaces via the theory of stable log maps. We then illustrate our method via the key example of Witten's $r$-spin class. In the subsequent articles, we will push the technique to the general situation. One novelty of our theory is that such a compactification admits two virtual cycles, a usual virtual cycle and a "reduced virtual cycle". A key result of this article is that the reduced virtual cycle in the $r$-spin case equals to the r-spin virtual cycle as defined using cosection localization by Chang--Li--Li. The reduced virtual cycle has the advantage of being $\mathbb{C}^*$-equivariant for a non-trivial $\mathbb{C}^*$-action. The localization formula has a variety of applications such as computing higher genus Gromov--Witten invariants of quintic threefolds and the class of the locus of holomorphic differentials

Topological recursion relations from Pixton's formula

For any genus g \leq 26, and for n \leq 3 in all genus, we prove that every degree-g polynomial in the psi-classes on Mbar_{g,n} can be expressed as a sum of tautological classes supported on the boundary with no kappa-classes. Such equations, which we refer to as topological recursion relations, can be used to deduce universal equations for the Gromov-Witten invariants of any target.Comment: 17 page