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 prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

This paper is devoted to determine the geometry of a class of smooth projective rational surfaces whose minimal models are the Hirzebruch ones; concretely, they are obtained as the blowup of a Hirzebruch surface at collinear points. Explicit descriptions of their effective monoids are given, and we present a decomposition for every effective class. Such decomposition is used to confirm the effectiveness of some divisor classes when the Riemann–Roch theorem does not give information about their effectiveness. Furthermore, we study the nef divisor classes on such surfaces. We provide an explicit description for their nef monoids, and, moreover, we present a decomposition for every nef class. On the other hand, we prove that these surfaces satisfy the anticanonical orthogonal property. As a consequence, the surfaces are Harbourne–Hirschowitz and their Cox rings are finitely generated. Finally, we prove that the complete linear system associated with any nef divisor is base-point-free; thus, the semi-ample and nef monoids coincide. The base field is assumed to be algebraically closed of arbitrary characteristic.

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

We extend Poincaré duality in étale cohomology from smooth schemes to regular ones. This is achieved via a formalism of trace maps for local complete intersection morphisms.

We propose a variation of the classical Hilbert scheme of points, the double nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss $K$-theoretic refinements à la Nekrasov–Okounkov.

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 study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21].

In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

We prove that a formula for the ‘pluricanonical’ double ramification cycle proposed by Janda, Pandharipande, Pixton, Zvonkine, and the second-named author is in fact the class of a cycle constructed geometrically by the first-named author. Our proof proceeds by a detailed explicit analysis of the deformation theory of the double ramification cycle, both to first and to higher order.

We study log $\mathscr {D}$-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.

Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree d associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at e real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu–Yott, and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson–Bloch conjecture on the injectivity of the higher Abel–Jacobi map, the generating series of special cycles of codimension er in the Chow group is a Hilbert–Siegel modular form of genus r and weight $1+n/2$. Our result is a generalization of Kudla’s modularity conjecture, solved by Yuan–Zhang–Zhang unconditionally when $e=1$.

We describe generators of disguised residual intersections in any commutative Noetherian ring. It is shown that, over Cohen–Macaulay rings, the disguised residual intersections and algebraic residual intersections are the same, for ideals with sliding depth. This coincidence provides structural results for algebraic residual intersections in a quite general setting. It is shown how the DG-algebra structure of Koszul homologies affects the determination of generators of residual intersections. It is shown that the Buchsbaum–Eisenbud family of complexes can be derived from the Koszul–Čech spectral sequence. This interpretation of Buchsbaum–Eisenbud families has a crucial rule to establish the above results.

We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo $p$ prevents the existence of an action without fixed points of certain finite $p$-groups. The case of base fields of characteristic $p$ is included. Counterexamples are systematically provided to test the sharpness of our results.

We prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold $G/B$ . In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of $G/B$ . By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold $G/P$ . We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern–Schwartz–MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a strong splayedness assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern–Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.