Let A = (a_1,…,a_n)∈ℤ^n be a sequence with sum k(2g-2+n). The double ramification cycle 𝖣𝖱_g(A) ∈𝖢𝖧^g(ℳ̅_g,n) is the virtual class of the locus of curves (C,p_1,…,p_n) where the line bundle (ω_C^log)^-k(∑ a_i p_i) is trivial. Although there has long been a formula for 𝖣𝖱_g(A) [JPPZ17], the exact dependence on A was unknown for a long time, though it was conjectured to be polynomial in A. A proof was announced in [JPPZ17], and Pixton gave a proof incorporating ideas of Zagier in [Pix23]. Here we present an alternative proof of the polynomiality of the double ramification cycle.
Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the projection onto one of the factors has birational image, we show that a transformation of its K-polynomial is Lorentzian. This partially answers a conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montano. As another application, we show that the h*-vector of a simplicially positive divisor on a matroid is a Macaulay vector, affirmatively answering a question of Speyer for a new infinite family of matroids.
We compute the constant of approximation for an arbitrary rational point on an arbitrary smooth cubic hypersurface X over a number field k, provided that there is a k-rational line somewhere on X. In the process, we verify the Coba conjecture for X.
We prove a generalisation of the Grothendieck-Riemann-Roch theorem, which is valid for any proper and flat morphism between noetherian and separated schemes of odd characteristic.
We consider arbitrary polarized variations of Hodge structure of weight two and h^2,0=1 over a non–singular complex algebraic curve S and analyze the boundary behaviour of the associated Kudla–Millson theta series using Schmid's theorems on degenerations of Hodge structure. This allows us to prove that this theta series is always integrable over S and to describe explicitly the non-holomorphic part of the Kudla–Millson generating series in terms of the mixed Hodge structures at infinity.
We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system π: M → B. Our main conjecture is that the Fourier-Mukai transform of sheaves of Kähler differentials, after restriction to the formal neighborhood of the zero section, are quantized by the Hodge modules arising in the decomposition theorem for π. For an integrable system, our formulation unifies the Fourier-Mukai calculation of the structure sheaf by Arinkin-Fedorov, the theorem of the higher direct images by Matsushita, and the "perverse = Hodge" identity by the second and the third authors. As evidence, we show that these Fourier-Mukai images are Cohen-Macaulay sheaves with middle-dimensional support on the relative Picard space, with support governed by the higher discriminants of the integrable system. We also prove the conjecture for smooth integrable systems and certain 2-dimensional families with nodal singular fibers. Finally, we sketch the proof when cuspidal fibers appear.
We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional variety. This question was studied by Gritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when the divisibility of the polarization is one. We generalize their results to higher dimension and divisibility. As a main result, for almost all dimensions $2n$ we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperk\"ahler varieties of $K3^{[n]}$ type is of general type.
We compute the Poincar\'e polynomials of the compactified Jacobians for plane curve singularities with Puiseaux exponents $(nd,md,md+1)$, and relate them to the combinatorics of $q,t$-Catalan numbers in the non-coprime case. We also confirm a conjecture of Cherednik and Danilenko for such curves.
Given a log smooth scheme $(X,D)$, and a log \'etale modification $(\tilde{X},\tilde{D}) \rightarrow (X,D)$, we relate the punctured Gromov-Witten theory of $(\tilde{X},\tilde{D})$ to the punctured Gromov-Witten theory of $(X,D)$, generalizing results of Abramovich and Wise in the non-punctured setting in "Birational invariance in log Gromov-Witten Theory". Using the main comparison results, we show a form of log \'etale invariance for the logarithmic mirror algebras and canonical scattering diagrams constructed in "Intrinsic Mirror Symmetry" and "The Canonical Wall Structure and Intrinsic Mirror Symmetry" respectively.
Using a construction by Yamamoto of tropical contractions, we construct a non-archimedean SYZ fibration on the Berkovich analytification of a class of maximally degenerate hypersurfaces in projective space. We furthermore prove that under a discrete symmetry assumption, the potential for the non-archimedean Calabi-Yau metric is constant along the fibers of the retraction. The proof uses the work of Li on the Fermat degeneration of hypersurfaces, and an explicit description of toric plurisubharmonic metrics on the hybrid space associated to a complex toric variety.
In this paper, we prove that the theta divisor of a smooth hyperelliptic curve has a natural and explicit embedded resolution of singularities using iterated blowups of Brill-Noether subvarieties. We also show that the Brill-Noether stratification of the hyperelliptic Jacobian is a Whitney stratification.
Given a map ϕ:X→ Y between F-analytic manifolds over a local field F of characteristic 0, we introduce an invariant ϵ_⋆(ϕ) which quantifies the integrability of pushforwards of smooth compactly supported measures by ϕ. We further define a local version ϵ_⋆(ϕ,x) near x∈ X. These invariants have a strong connection to the singularities of ϕ. When Y is one-dimensional, we give an explicit formula for ϵ_⋆(ϕ,x), and show it is asymptotically equivalent to other known singularity invariants such as the F-log-canonical threshold lct_F(ϕ-ϕ(x);x) at x. In the general case, we show that ϵ_⋆(ϕ,x) is bounded from below by the F-log-canonical threshold λ=lct_F(𝒥_ϕ;x) of the Jacobian ideal 𝒥_ϕ near x. If Y= X, equality is attained. If Y< X, the inequality can be strict; however, for F=ℂ, we establish the upper bound ϵ_⋆(ϕ,x)≤λ/(1-λ), whenever λ<1. Finally, we specialize to polynomial maps φ:X→ Y between smooth algebraic ℚ-varieties X and Y. We geometrically characterize the condition that ϵ_⋆(φ_F)=∞ over a large family of local fields, by showing it is equivalent to φ being flat with fibers of semi-log-canonical singularities.
Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higherdimensional analogues of K3 surfaces, Y. André has verified the Shafarevich conjecture for hyperKähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyperKähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach is a uniform Kuga–Satake map, inspired by She’s work, and we study its arithmetic properties, which are of independent interest.
Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and that a general cubic surface arises 42120 times as a hyperplane section of a general cubic 3-fold.
Let $\mathcal{X} \to Y$ be a birational map from a smooth Artin stack to a (possibly singular) variety. We prove a change of variables formula that relates motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. With a view toward the study of stringy Hodge numbers, this change of variables formula leads to a new notion of crepantness for the map $\mathcal{X} \to Y$ that coincides with the usual notion in the special case that $\mathcal{X}$ is a scheme.
Obus-Wewers and Pop recently resolved a long-standing conjecture by Oort that says: every cyclic cover of a curve in characteristic p>0 lifts to characteristic zero. Saïdi further asks whether these covers are also "liftable in towers". We prove that the answer for the equal-characteristic version of this question is affirmative. Our proof employs the Hurwitz tree technique and the tools developed by Obus-Wewers.
