We define the tropical Tevelev degrees, ${\mathsf{Tev}}_g^{\mathtt{trop}}$, as the degree of a natural finite morphism between certain tropical moduli spaces, in analogy to the algebraic case. We develop an explicit combinatorial construction that computes ${\mathsf{Tev}}_g^{\mathtt{trop}} = 2^g$. We prove that these tropical enumerative invariants agree with their algebraic counterparts, giving an independent tropical computation of the algebraic degrees ${\mathsf{Tev}}_g$.

We present a formula for the Poincaré dual in the flag manifold of the equivariant fundamental class of any regular nilpotent or regular semisimple Hessenberg variety as a polynomial in terms of certain Chern classes. We then develop a type-independent proof of the Giambelli formula for the Peterson variety and use this formula to compute the intersection multiplicity of a Peterson variety with an opposite Schubert variety corresponding to a Coxeter word. Finally, we develop an equivariant Chevalley formula for the cap product of a divisor class with a fundamental class, and a dual Monk rule, for the Peterson variety.

We describe the behavior of a free reduced plane projective curve with respect to the addition, respectively, deletion, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

We study the relationship between the enumerative geometry of rational curves in local geometries and various versions of maximal contact logarithmic curve counts. Our approach is via quasimap theory, and we show versions of the [vGGR19] local/logarithmic correspondence for quasimaps, and in particular for normal crossings settings, where the Gromov-Witten theoretic formulation of the correspondence fails. The results suggest a link between different formulations of relative Gromov-Witten theory for simple normal crossings divisors via the mirror map. The main results follow from a rank reduction strategy, together with a new degeneration formula for quasimaps.

A few years ago, G. Oberdieck conjectured a multiple cover formula that determines the number of curves of fixed genus and degree passing through a configuration of points in an abelian surface. This formula was proved by the author using tropical techniques and Nishinou’s correspondence theorem. Using the same techniques, we give a much shorter proof of the multiple cover formula for point insertions, relying on the same geometrical idea, but avoiding any kind of tropical enumeration.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.

The blow-up of the anticanonical base point on a del Pezzo surface S of degree 1 gives rise to a rational elliptic surface $\mathscr {E}$ with only irreducible fibers. The sections of minimal height of $\mathscr {E}$ are in correspondence with the $240$ exceptional curves on S. A natural question arises when studying the configuration of these curves: if a point on S is contained in “many” exceptional curves, is it torsion on its fiber on $\mathscr {E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if “many” equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if ‘many’ equals $9$ or more. Moreover, we give counterexamples where a non-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.

Given any smooth germ of a 3-fold flopping contraction, we first give a combinatorial characterisation of which Gopakumar–Vafa (GV) invariants are non-zero, by prescribing multiplicities to the walls in the movable cone. On the Gromov–Witten (GW) side, this allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of Iyama and the second author and, moreover, that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix that controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes; (2) a closed formula, which describes the change in dimension of the contraction algebra under flop; and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

We provide an explicit formula for all primary genus-zero $r$-spin invariants. Our formula is piecewise polynomial in the monodromies at each marked point and in $r$. To deduce the structure of these invariants, we use a tropical realisation of the corresponding cohomological field theories. We observe that the collection of all Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) relations is equivalent to the relations deduced from the fact that genus-zero tropical CohFT cycles are balanced.

We study the enumerativity of Gromov–Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree $d \leq (n+3)/3$ in ${\mathbb P}^n$.

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.

We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

We address Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin.

These techniques seem also suitable to deal with three orthogonal generalisations: (1) the extension to the r-hyperelliptic locus; (2) the extension to an arbitrary number of non-Weierstrass pairs of points; (3) the extension to multiple descendants.

In this paper, we prove a series of identities of the quasi-map K-theoretical I-functions with level structure between the Grassmannian and its dual Grassmannian. Those identities prove the quantum K-theory version mutation conjecture stated in [13]. Here we find an interval of levels within which two I-functions are the same, and on the boundary of that interval, two I-functions intertwine. We call this phenomenon the level correspondence in Grassmann duality.

We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in $\overline{{\mathcal{M}}}_{g,n}$ which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors.

In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus $g$ curves are of pure codimension $g$ in $\overline{{\mathcal{M}}}_{g,n}$ . In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors.

As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of rational functions (up to automorphism of the target) on the projective line with ramification to order ei at general points Pi, in the case that all ei are less than the characteristic. Unlike the case of characteristic 0, the answer is not given by Schubert calculus, nor is the number of maps always finite for distinct Pi, even in the tamely ramified case. However, finiteness for general Pi, obtained by exploiting the relationship to branched covers, is a key part of the argument.

We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation ${{\Lambda }^{k}}{{\mathbb{C}}^{n}}$