The tautological ℚ-subalgebra 𝖱^*(𝒜_g) ⊂𝖢𝖧^*(𝒜_g) of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical ℚ-linear projection operator 𝗍𝖺𝗎𝗍: 𝖢𝖧^*(𝒜_g) →𝖱^*(𝒜_g). We present here new calculations of intersection products of the Torelli locus in 𝒜_g with the product loci 𝒜_r×𝒜_g-r→𝒜_g for r≤ 3. The results suggest that 𝗍𝖺𝗎𝗍 is a ℚ-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property. Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps. We define tautological projections of cycles on the fiber products 𝒳_g^s →𝒜_g of the universal family. We compute these projections for a class of product cycles on 𝒳_g^s in terms of a determinant involving the universal theta divisors and Poincaré classes. Using Abel-Jacobi pullbacks of product cycles on 𝒳_g^s and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings 𝖱^*(ℳ^ct_g,n). In particular, we construct a nontrivial element of the Gorenstein kernel of 𝖱^5(ℳ_5,2^ct).

All reduced descendent Gromov-Witten invariants of K3 and abelian surfaces in primitive curve classes can be calculated by the methods of . To handle the imprimitive curve classes, a multiple cover formula was conjectured in for K3 surfaces and in for abelian surfaces. We prove here that both descendent multiple cover formulas are implied by the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The implication is proven by showing that the multiple cover formula for S can be recast as a property of an appropriate localization vertex for the relative 3-fold Gromov-Witten theory of (S×ℙ^1/S_0 ∪ S_∞). The families GW/PT correspondence then transfers the multiple cover formula from the Gromov-Witten side to the stable pairs side where the formula is proven geometrically by studying cosections and applying universality properties. Along the way, we prove a DT/PT correspondence for the reduced theories of (S×ℙ^1/S_0 ∪ S_∞) using the wallcrossing techniques of Kuhn-Liu-Thimm .

Pagani and Tommasi have introduced a class of smoothable fine compactified Jacobians $\overline{\mathcal{J}}_{g,n}^d(σ)\rightarrow \overline{\mathcal{M}}_{g,n}$ over the moduli space of stable curves, depending nontrivially on the degree $d$ and the choice of a stability condition $σ$. A theorem of Migliorini-Shende-Viviani implies that the cohomology of $\overline{\mathcal{J}}_{g,n}^d(σ)$ is independent of $d$ and $σ$, a statement which is quite unexpected from the point of view of the boundary geometry of these spaces. We reprove this independence statement using a direct combinatorial argument, summing up contributions of individual strata. The Appendix includes a result by J. Feusi characterizing when $\mathcal{J}_{g,n}^d$ and $\mathcal{J}_{g,n}^{d'}$ are $S_n$-equivariantly isomorphic over $\mathcal{M}_{g,n}$, and a result by Q. Yin showing that $[\mathcal{J}^d_g]$ and $[\mathcal{J}^{d'}_g]$ are not always equal in $K_0(\text{Var}_{\mathbb{C}})$.

Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus $g$ maps of fixed complex structure in a given curve class $\beta$ through $n$ general points of a target variety $X$. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class $\beta$. A simple speculation is that for all Fano $X$, the virtual Tevelev degrees are enumerative for sufficiently large $\beta$. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension). As an application, we prove a new result on the existence of very free curves of low degree on hypersurfaces in positive characteristic.

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme $\mathsf{Hilb}^n(\mathbb{C}^2)$ of points in the plane. The fundamental 1-point invariant (with a divisor insertion) is calculated using a correspondence with the families local curve Gromov-Witten theory over the moduli space $\overline{\mathcal{M}}_{1,1}$. The answer exactly matches a parallel calculation related to the Noether-Lefschetz geometry of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties. As a consequence, we prove that the associated cycle classes satisfy a homomorphism property for the projection operator on $\mathsf{CH}^*(\mathcal{A}_g)$. The fundamental 1-point invariant determines the full genus 1 Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ modulo a nondegeneracy conjecture about the quantum cohomology. A table of calculations is given.

We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras are infinite dimensional, a connection to polyhedral combinatorics via a new theory of homological piecewise polynomials allows an effective study. A complete calculation is given in genus 0 via the algebra of piecewise polynomials on the cone stack of the associated Artin fan (lifting Keel's presentation of the Chow ring of M0,n). Counterexamples to the simplest generalizations in genus 1 are presented. We show, however, that the structure of the log tautological rings is determined by the complete knowledge of all relations in the standard tautological rings of the moduli spaces of curves. In particular, Pixton's conjecture concerning relations in the standard tautological rings lifts to a complete conjecture for relations in the log tautological rings of the moduli spaces of curves. Several open questions are discussed. We develop the entire theory of logarithmic tautological classes in the context of arbitrary smooth normal crossings pairs (X, D) with explicit formulas for intersection products. As a special case, we give an explicit set of additive generators of the full logarithmic Chow ring of (X, D) in terms of Chow classes on the strata of X and piecewise polynomials on the cone stack. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Algebraic geometry has grown into a broad subject, with many different streams often advancing quite independently of each other. Nonetheless, important advances have often come from visionary applications of ideas in one part of the subject to another. This workshop brought together leaders and future leaders in different areas of the subject, centered on geometric methods or geometric problems. It also brought together groups from different regions of the globe, in order to bridge communities of different sorts, and help new ideas quickly spread throughout algebraic geometry. Some of the best freshly-minted algebraic geometers were deliberately invited, so that they could meet their peers from around the world and learn about different perspectives on the subject.

The tautological Chow ring of the moduli space 𝒜_g of principally polarized abelian varieties of dimension g was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from 𝒜_g to the moduli space ℳ_g^ct of genus g of curves of compact type, we prove that the product class [𝒜_1×𝒜_5]∈𝖢𝖧^5( 𝒜_6) is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the tautological ring 𝖱^*(ℳ_6^ct) in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring 𝖱^*(ℳ_6^ct) has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of [𝒜_1×𝒜_5] . More generally, the Torelli pullback of the difference between [𝒜_1×𝒜_g-1] and its tautological projection always lies in the Gorenstein kernel of 𝖱^*(ℳ_g^ct) . The product map 𝒜_1×𝒜_g-1→𝒜_g is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer’s tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all g are presented.

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme 𝖧𝗂𝗅𝖻^n(ℂ^2) of points in the plane. The fundamental 1-point invariant (with a divisor insertion) is calculated using a correspondence with the families local curve Gromov-Witten theory over the moduli space ℳ_1,1. The answer exactly matches a parallel calculation related to the Noether-Lefschetz geometry of the moduli space 𝒜_g of principally polarized abelian varieties. As a consequence, we prove that the associated cycle classes satisfy a homomorphism property for the projection operator on 𝖢𝖧^*(𝒜_g). The fundamental 1-point invariant determines the full genus 1 Gromov-Witten theory of 𝖧𝗂𝗅𝖻^n(ℂ^2) modulo a nondegeneracy conjecture about the quantum cohomology. A table of calculations is given.

Relationships between moduli spaces of curves and sheaves on 3-folds are presented starting with the Gromov-Witten/Donaldson-Thomas correspondence proposed more than 20 years ago with D. Maulik, N. Nekrasov, and A. Okounkov. The descendent and relative correspondences as developed with A. Pixton in the context of stable pairs led to the proof of the correspondence for the Calabi-Yau quintic 3-fold. More recently, the study of correspondences in families has played an important role in connection with other basic moduli problems in algebraic geometry. The full conjectural framework is presented here in the context of families of 3-folds. This article accompanies my lecture at the ICBS in July 2024.

We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties 𝒜_g: every cycle class decomposes canonically as a sum of a tautological and a non-tautological part. The main new result required for the definition of the projection operator is the vanishing of the top Chern class of the Hodge bundle over the boundary 𝒜̅_g∖𝒜_g of any toroidal compactification 𝒜̅_g of the moduli space 𝒜_g. We prove the vanishing by a careful study of residues in the boundary geometry. The existence of the projection operator raises many natural questions about cycles on 𝒜_g. We calculate the projections of all product cycles 𝒜_g_1×…×𝒜_g_ℓ in terms of Schur determinants, discuss Faber's earlier calculations related to the Torelli locus, and state several open questions. The Appendix contains a conjecture about the projection of the locus of abelian varieties with real multiplication.

Let $A=(a_1,\ldots, a_n)$ be a vector of integers which sum to $k(2g-2+n)$. The double ramification cycle $\mathsf{DR}_{g,A}\in \mathsf{CH}^g(\mathcal{M}_{g,n})$ on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves $(C,x_1,\ldots,x_n)$ satisfying $$\mathcal{O}_C\Big(\sum_{i=1}^n a_i x_i\Big) \, \simeq\, \big(\omega^{\mathsf{log}}_{C}\big)^k\, .$$ The Abel-Jacobi construction requires log blow-ups of $\mathcal{M}_{g,n}$ to resolve the indeterminacies of the Abel-Jacobi map. Holmes has shown that $\mathsf{DR}_{g,A}$ admits a canonical lift $\mathsf{logDR}_{g,A} \in \mathsf{logCH}^g(\mathcal{M}_{g,n})$ to the logarithmic Chow ring, which is the limit of the intersection theories of all such blow-ups. The main result of the paper is an explicit formula for $\mathsf{logDR}_{g,A}$ which lifts Pixton's formula for $\mathsf{DR}_{g,A}$. The central idea is to study the universal Jacobian over the moduli space of curves (following Caporaso, Kass-Pagani, and Abreu-Pacini) for certain stability conditions. Using the criterion of Holmes-Schwarz, the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz applied to the universal line bundle determines the logarithmic double ramification cycle. The resulting formula, written in the language of piecewise polynomials, depends upon the stability condition (and admits a wall-crossing study). Several examples of logarithmic and higher double ramification cycles are computed.

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the�-function of an open KdV hierar- chy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the the- ory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a the- ory of higher genus open descendent integrals. As a result, we ob- tain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves.

We study the Chow ring with rational coefficients of the moduli space F 2 \mathcal{F}_{2} of quasi-polarized K3 surfaces of degree 2. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is A 17 ⁢ ( F 2 ) ≅ Q \mathsf{A}^{17}(\mathcal{F}_{2})\cong{\mathbb{Q}} . We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into A 17 ⁢ ( F 2 ) \mathsf{A}^{17}(\mathcal{F}_{2}) . The kernel of the pairing is a 1-dimensional subspace of A 9 ⁢ ( F 2 ) \mathsf{A}^{9}(\mathcal{F}_{2}) which we calculate explicitly. In the appendix, we revisit Kirwan–Lee’s calculation of the Poincaré polynomial of F 2 \mathcal{F}_{2} .

We construct irrational irreducible components of the Hilbert scheme of points of affine n-dimensional space, for n at least 12. We start with irrational components of the Hilbert scheme of curves in P^3 and use methods developed by Jelisiejew to relate these to irreducible components of the Hilbert schemes of points of A^n. The result solves Problem XX of [J. Jelisiejew, Open problems in deformations of Artinian algebras, Hilbert schemes and around, arXiv:2307.08777, 2023].

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 provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov-Witten theory are stated in complete generality and are of independent interest.

Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel-Jacobi map, we construct a universal twisted double ramification cycle $\mathsf{DR}^{\mathsf{op}}_{g,A}$ as an operational Chow class on the Picard stack $\mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus $g$ curves carrying a degree $d$ line bundle. The method of construction follows the log (and b-Chow) approach to the standard double ramification cycle with canonical twists on the moduli space of curves [arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471]. Our main result is a calculation of $\mathsf{DR}^{\mathsf{op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton's formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [arXiv:1812.10136]. The formula on the Picard stack is obtained from [arXiv:1812.10136] for target varieties $\mathbb{CP}^n$ in the limit $n \rightarrow \infty$. The result may be viewed as a universal calculation in Abel-Jacobi theory. As a consequence of the calculation of $\mathsf{DR}^{\mathsf{op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $\overline{\mathcal{M}}_{g,n}$ are exactly given by Pixton's formula (as conjectured in the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]). The comparison result of fundamental classes proven in [arXiv:1909.11981] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $\mathfrak{Pic}_{g,n,d}$ associated to Pixton's formula.

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 generalization). We find exact solutions which specialize to Tevelev's formula in his cases and connect to the projective geometry of lines and Castelnuovo's classical count of linear series in other cases. For almost all values, the calculation of the two parameter generalization of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

We define stationary descendent integrals on the moduli space of stable maps from disks to (CP1,RP1). We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus 0 disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.