Papers

We classify all open substacks admitting a proper good moduli space inside the stack $\mathcal{G}_{1,n}$ of log-canonically polarized Gorenstein curves of genus one with $n$ distinct marked points: they are exactly the stacks $\overline{\mathcal{M}}_{1,n}(c)$ defined by Bozlee–Kuo–Neff. Restricting to the case where $c$ is pure of level $m$, we then give a numerical criterion for the projectivity of the good moduli spaces: these turn out to be asymptotically almost never projective as $n$ tends to infinity.

Let $S_E$ be the surface in Xiao's degree-four family identified by Polizzi as a smooth divisor in $E(3)$. Starting from Polizzi's branch model and Xiao's classification of the thirteen singular fibers, we derive the local spherical braids and their Picard–Lefschetz lifts, including the colored $3+3$ partitions at the seven reducible fibers. We identify the normalization of each bisection with $E/\{\pm1\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove $\pi_1(S_E)\cong\mathbb Z^2$, compute the elementary-divisor-$4$ Albanese kernel of a regular genus-two fiber, and show that the six nonseparating Picard–Lefschetz transformations represent the six cusps of $\Gamma(4)$. We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower $b_1$ below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic $\mathbb CP^2\#7\overline{\mathbb CP}^{\,2}$.

We develop a deformation-theoretic framework for algebraic superstacks and apply it to the construction of perfect obstruction theories on moduli spaces of stable supermaps. In order to achieve this we employ the tools and concepts from derived algebraic geometry, extending them to the supergeometric setting. Our main technical contributions are (i) the notions of derived superstacks, (ii) their cotangent supercomplexes and (iii) perfect obstruction theories for algebraic superstacks. We prove that a quasi-smooth derived enhancement of an algebraic superstack always yields a perfect obstruction theory: this generalizes a well-known principle from classical deformation theory. We then turn to the case of the Deligne–Mumford superstack $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ of stable supermaps of fixed genus, homology class and numbers of Neveu–Schwarz and Ramond–Ramond punctures, with target a smooth projective superscheme $\boldsymbol{Y}$. We exhibit a natural derived enhancement $\mathbb{R} \boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ and compute its cotangent supercomplex; we prove that the induced obstruction theory on $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ is perfect. Finally, we compute its virtual dimension by super Grothendieck–Riemann–Roch. The resulting formula gives, in particular, a deformation-theoretic proof of the virtual dimension formula previously conjectured for stable supermaps and specializes to the usual virtual dimension of the classical moduli stack of stable maps.

We study the Gieseker moduli space $\mathcal{M}_{X_0}$ of stable rank-$2$, degree-$1$ vector bundles on an irreducible curve $X_0$ with one node. Its normalization is a two-step blow-up of the moduli space of generalized parabolic bundles on the normalization $\tilde X_0$, and $\mathcal{M}_{X_0}$ is recovered by gluing along a natural involution on the boundary. From this we compute $\operatorname{Pic}(\mathcal{M}_{X_0})$ and, via a Harder–Narasimhan stratification, the virtual Poincaré polynomial of $\mathcal{M}_{X_0}$. Since the normalization morphism is small, the intersection cohomology of $\mathcal{M}_{X_0}$ equals the ordinary cohomology of the smooth normalization, which gives a closed formula in terms of the arithmetic genus.

We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + $\varepsilon$)n due to (Landsberg and Michalek 2025). We also prove that the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n-1. When n is odd, we prove that this bound is achieved by Koszul flattenings. When n is even, a result of (Landsberg 2015) shows that Koszul flattenings can achieve 2n-2, leaving an open gap of size one. Our 2n-1 bound improves the best known 6n-4 linear flattening barrier due to (Garg et al. 2019) and (Buczyński 2026). Combined, these results show that our 3n-o(n) construction, as well as the construction of Landsberg and Michalek, provide explicit examples of tensors with higher border rank than any linear flattening can achieve.

Plane Cremona maps of degree four are divided into two different classes according to their homaloidal type: either they are de Jonquières, that is, they have a base point of multiplicity 3, or they are not de Jonquières. The maps of the former class have length one, while the maps of the latter class have length two, according to the definition of length of a plane Cremona map given by Blanc and Furter. In this paper, we classify plane Cremona maps of degree four and length two, up to the action of automorphisms of the projective plane on the left-hand side and on the right-hand side of the map. This classification also allows us to compute the ordinary quadratic length of all such maps. In particular, we show that this ordinary quadratic length is at most 7 and we classify those maps which attain this upper bound.

A recognizer that compares rotation-invariant descriptors sees a surface only up to the fiber of the descriptor. We measure this fiber by its radius in the orbit distance from the enrolled surface. A large radius admits decoys, that is, distant shapes that pass the matcher. A small radius discloses the enrolled shape to anyone who captures the stored value. For star-shaped surfaces truncated to spherical harmonics of degree at most $L$, with $n$ coefficients, a descriptor of generic rank $r$ has generic fibers of dimension $n-3-r$ modulo rotations. The standard pool of band powers, even bispectra, and three invariants of the degree-three band therefore admits decoy families of dimension $5$, $13$, $20$ at $L=4,6,8$. Its rank first reaches $n-3$ at $L=16$, and a mirror decoy remains at every $L$. The odd bispectra remove the mirror decoy generically for $L \geq 4$. Yet at fixed mean radius the same pool determines the enclosed volume exactly, and it does not determine whether a surface meets a clearance requirement. We certify two cases by exact and interval arithmetic. At $L=6$ a decoy matches all $32$ invariants to relative precision $2 \cdot 10^{-18}$ at orbit distance at least $0.87$ times the norm of the enrolled tuple. For the radar shape model of asteroid (101955) Bennu, the pool recovers the modeled volume, misses the handedness, and leaves the keep-out radius uncertain by more than $7 \, \mathrm{m}$.

We introduce new techniques to study mixed-characteristic special loci in moduli spaces and give several applications, focusing in particular on the moduli space $\mathsf{M}_g$ of curves. These applications include Zariski density results for split Jacobians, non-density results for supersingular loci, and lifting results for algebraic cycles, all under appropriate numerical conditions. Our methods generalize to more-or-less arbitrary moduli spaces mapping to Hodge-type Shimura varieties (and even some settings beyond the Shimura case), and relate to questions raised by Oort and de Jong.

Let $L$ be a number field containing a CM field $K$. We compute an explicit current representing the $\mathrm{GL}_{\mathcal{O}_K}(\mathcal{O}_L)$-equivariant algebraic Eisenstein-Kronecker classes constructed by Kings-Sprang. In the Bianchi case, where $K$ is imaginary quadratic and $[L:K]=2$, this yields a purely algebraic construction of Eisenstein cohomology classes, even though the associated locally symmetric space is not an algebraic variety. We express their constant terms at the cusp in terms of partial Hecke $L$-values of $K$.

We construct distinct high-dimensional mean-zero finite range lattice random walks having pairwise-equal return probabilities for all step counts. The same examples provide pairwise-distinct shapes of discretizations of the standard Laplacian with pairwise-equal density state functions. The main contribution is a reconstruction theorem: to a colored trivalent graph one associates a quantum Clebsch–Gordan polytope, and this association is a full functor, in particular from the polytope one can uniquely recover the original graph. These polytopes appear as moment polytopes of toric degenerations of character varieties (moduli spaces of rank-2 bundles on curves), yielding a combinatorial non-abelian Torelli theorem. In symplectic geometry, the reconstruction theorem implies that monotone Lagrangian tori on odd character varieties associated with these degenerations are pairwise non-Hamiltonian isotopic. These results, and some of the applications, arose from the study of mirror symmetry for moduli spaces of vector bundles, and of the related Laurent phenomenon for mutations of graph potentials.

Let $I$ be the defining ideal of a general set of $s$ points in projective $N$-space over an algebraically closed field. Harbourne and Huneke conjectured the strengthened symbolic containment $$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$ for every $r\ge 1$. For general points this was known in low dimension and for small numbers of points, while in arbitrary dimension and arbitrary cardinality the best general result was the stable version, valid on one dense Zariski-open set for all sufficiently large $r$. We show that the recent proof of Demailly's conjecture for arbitrary finite point sets by Hà and Sivakumar removes the stability threshold. The key observation is that the strict generic Waldschmidt estimate of Bisui and Nguyẽn allows one to select a single finite symbolic level $m_0$. Equality of the corresponding interpolation number is a Zariski-open condition. Demailly's inequality then propagates this one finite condition to lower bounds for every symbolic power, and a standard regularity criterion converts those bounds into the Harbourne–Huneke containment for every $r$. Consequently, for every $N\ge2$ and every $s\ge1$, one dense Zariski-open family of $s$-point configurations in $\mathbb{P}^N$ satisfies the Harbourne–Huneke containment simultaneously for all $r\ge 1$. The argument also clarifies a quantifier issue in the earlier literature: before the full Demailly theorem, estimates on the generic fibre naturally yielded either all $r$ on a very general set, or one open set only for $r\gg 0$. The new theorem replaces infinitely many symbolic conditions by a single finite interpolation condition.

In this paper, we prove that the combination of dense klt type and $\mathcal{D}$-simplicity property establishes the klt type property.

We study Hodge–Tjurina numbers and the Tjurina subspectrum of isolated hypersurface singularities. We relate these invariants to Hodge ideals and microlocal $V$-filtrations and investigate their behavior under Thom–Sebastiani sums. For separated sums, we obtain a filtered defect formula and, in particular, give an affirmative answer to the question raised by Almirón concerning the Thom–Sebastiani property of the Tjurina subspectrum: it holds if and only if the Tjurina number is multiplicative. We also discuss rational sampling, spectral reconstruction, symmetry bounds, and examples illustrating the information captured and lost by integer Hodge–Tjurina data.

We show that any integral cubic form in ten or more variables has a nontrivial integer zero. Like earlier authors on this topic, we use geometric methods to produce a zero in some cases, and the circle method in all other cases. New ingredients are required on both fronts: in the first case, bounds on the dimension of singular loci and incidence varieties, and in the second case, a stratified geometric sieve, together with stratification results for point counts over finite fields and a cube-root Hensel lifting analysis.

We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.

We study the diagonal Specht ideals $I_\lambda\subseteq\mathcal{R}_{m,n}:=\mathbb{C}[\mathbf{x}_1,\dots,\mathbf{x}_m]$, $\mathbf{x}_r=(x_{r,1},\dots,x_{r,n})$, generated by the $\lambda$-isotypic component of $\mathcal{R}_{m,n}$ for the diagonal action of $S_n$. We characterize the set of zeros of $I_\lambda$ as $V_\lambda=\bigcup_{\mu\not\trianglelefteq\lambda}H_\mu$ and prove that $\lambda\mapsto V_\lambda$ is an isomorphism of posets from $(\mathcal{P}_n,\trianglelefteq)$ to $(\{V_\lambda:\lambda\in \mathcal{P}_n\},\supseteq)$ for all $m$, while $\lambda\mapsto I_\lambda$ is an isomorphism of posets from $(\mathcal{P}_n,\trianglelefteq)$ to $(\{I_\lambda:\lambda\in \mathcal{P}_n\},\subseteq)$ if and only if $m=1$ or $n\leq3$. Unlike the case $m=1$, where Specht ideals are always radical, radicality in the diagonal setting depends on $\lambda$ and $m$. We prove radicality for hook partitions of length at most three by providing explicit Gröbner bases, and we develop three criteria for non-radicality via content, multidegree, and total degree, showing in particular that $I_\lambda$ is not radical for any non-hook partition $\lambda$ when $m\geq\operatorname{len}(\lambda)$. Using the $\operatorname{GL}_m(\mathbb{C})$-action on $\mathcal{R}_{m,n}$, we show that $I_\lambda$ is radical for all $m$ if and only if it is radical for $m=n$. Finally, for $m\geq2$, we prove that $\mathcal{R}_{m,n}/I_\lambda$ is Cohen-Macaulay if and only if $\lambda=(n)$ or $\lambda=(n-1,1)$, and the same is true for $\mathcal{R}_{m,n}/\operatorname{rad}(I_\lambda)$.

Assuming projective log resolutions, we construct semiample perturbations over arbitrary fields that preserve all log canonical places. As an application, for divisor classes with finitely generated section rings, we characterize the existence of effective divisors yielding lc or klt pairs in terms of asymptotic orders of vanishing. We also derive minimal model and semiampleness results for generalized threefolds over $F$-finite fields of characteristic $p>5$.

We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over $\mathbb{Q}$. The proof establishes semisimplicity of $\ell$-adic cohomology by eliminating the unipotent radical of the algebraic monodromy group of total cohomology. We also prove the weight-monodromy conjecture for hyper-Kähler varieties over $p$-adic fields. For hyper-Kähler varieties over number fields with $b_2\geq 4$, we establish, after suitable finite extensions, strong compatibility of the associated Weil-Deligne representations valued in Mumford-Tate groups and its integral refinement away from finitely many primes.

This article is the third of a series of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In the second one, we generalized a topos-theoretic version of this transform, due to Oyama. Let $W$ be the ring of Witt vectors of a perfect field of positive characteristic $p$ and equip $\operatorname{Spf}W$ with the trivial log structure. Let $\mathfrak{S}$ be an fs log $p$-adic formal scheme, locally of finite type and log flat over $\operatorname{Spf}W$ and denote its special fiber by $S.$ For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we constructed crystalline-like ringed topoi $\mathcal{E}'$ and $\underline{\mathcal{E}}$ and subcategories of crystals of quasi-coherent modules $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi $\underline{\mathcal{E}} \rightarrow \mathcal{E}'.$ We proved that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and induces a fully faithful functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}}.$ Since the Frobenius morphism is not, in general, flat in the log smooth setting, it is not clear that this functor is essentially surjective. To address this issue, we refine the topoi and crystals mentioned above by endowing them with an indexed structure, inspired by Lorenzon's indexed extension of Cartier descent to smooth logarithmic schemes. Using the Azumaya property of the indexed algebra of logarithmic differential operators, we then obtain an equivalence between the corresponding categories of indexed crystals, thereby generalizing the Cartier transform.

Essential dimension counts the parameters needed to define a torsor, but it can drop sharply over a field extension, and it does not say how large an extension is needed. For a $G$-torsor $\tau$ over a field $F\supseteq k$ we study the finite-degree profile $\edd d_k(\tau)=\min\{\ed_k(\tau\otimes_FF'):[F':F]\le d\}$, a variant of invariants of Farb–Kisin–Wolfson and Farb–Wolfson, and its jump degrees $d_j(\tau)$, the least degree of an extension over which $\tau$ needs only $j$ parameters. The profile is geometric over every field: $\edd d_k(\tau)\le j$ iff some generically free $G$-variety of dimension $\le j$ has a closed point of degree $\le d$ in the free locus of its twist by $\tau$. We prove a Sylow divisibility theorem for jump degrees and a valuation-theoretic rank–index theorem, $\edd d_k(\tgen)\ge\min\{\rk B:B\le A,\ [A:B]\le d\}$ for abelian $A$ of order prime to $\operatorname{char}k$, which determines the profile of every abelian group over a field with enough roots of unity, for instance $n-\lfloor\log_2d\rfloor$ for $(\Z/2)^n$. For $\Z/p$ in characteristic $0$ we show $\lceil m/2\rceil\le d_1\le[k(\zeta_p+\zeta_p^{-1}):k]$, $m=[k(\zeta_p):k]$, with equality for $m$ even. For $S_5$ in characteristic $0$ we prove $d_1=4$, so Klein's reduction of the generic quintic is optimal; the lower bound comes from Abel–Jacobi rigidity, not from cohomological invariants. For $n\ge6$, $6\mid d_1(S_n)$.

We provide examples of smooth sextic double solids with potentially dense sets of rational points, over any field of characteristic zero.

We prove that the set of real $\mathrm{SL}_n$-opers forms a discrete subset of the moduli space of irreducible flat connections.

We prove that every smooth quartic double $n$-fold, $n\ge2$, over an algebraically closed field of characteristic zero is algebraically elliptic in Gromov's sense. In particular, every smooth quartic double solid is algebraically elliptic, which answers Question~4.28 in [M. Zaidenberg, Algebraic Gromov ellipticity: a brief survey, Taiwanese J. Math. 29 (2025), no. 6, 1681-1705]. We also prove that removing any closed subset of codimension at least two preserves algebraic ellipticity for smooth quartic double spaces and smooth cubic hypersurfaces of dimension at least two.

We study a representative two-loop family of worldline quantum field theory integrals relevant for classical gravitational scattering with a massive mediator and investigate the geometric structure of its maximal cut. We find that, unlike the massless case, the massive loops already give rise to an elliptic maximal cut at two loops, which degenerates into a nodal curve in the massless limit and in the static limit. In the global Baikov representation, two successive integrations reduce the maximal cut to a period of an elliptic curve. We further establish five primary consequences of this study: (a) The integration-by-parts method generates five master integrals, which, under reflection, split into a three-dimensional elliptic block and a two-dimensional polylog block. We derive the differential equation in two ways, one by Laporta reduction and subsequently by intersection theory in two cocycle bases, and construct the rational map between them. (b) The period matrices in arbitrary dimension have been computed in terms of Appell and Beta functions. Furthermore, we derive twisted Riemann period relations that yield nine quadratic identities among them. (c) We study the global monodromy relation of the differential system that satisfies Riemann's bilinear relation. (d) We explicitly derive the function space of the iterated integrals in terms of elliptic multiple polylogs. (e) In impact-parameter space, the Fourier-transformed cut is a Bessel-twisted differential on the elliptic curve, which we show has extra involution symmetries. Thus, the parity of the differential further decides which quotient - rational or elliptic its periods live on. All results presented concern the maximal cut and, therefore, the homogeneous solutions, but they open an important pathway to tackling the full massive two-loop integral.

Consider a system of equations $A(x)\cdot F(x) = B(x)$ on a subset $X$ of $\mathbb R^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a $\mathcal C^m$ vector-valued function $f$ on $\mathbb R^n$, and ask whether we can find a $\mathcal C^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\dim X = 1$. The methods are based on an article of Fefferman and Luli for $X \subset \mathbb R^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\dim X =1$, or in the case of the semialgebraic Whitney extension problem in $\mathbb R^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\mathbb R$.

The Schubert compactification of a linear space of matrices $L \subseteq \textrm{Mat}_{n\times m}$ is a natural compactification of $L$ in the Grassmannian $\textrm{Gr}(n, n+m)$. This construction generalises classical Schubert varieties and matroid Schubert varieties, and it turns invariants of linear matrix spaces into projective and intersection-theoretic invariants of the associated variety. We develop this geometry completely in the case of matrix pencils, that is $\dim L=2$. The resulting Schubert surfaces are singular analogues of rational surfaces obtained as blow-ups of $\mathbb{P}^2$. We show that they are normal with quotient singularities, compute their degree, cohomology ring, and describe the distinguished rational curves and their intersection pairing in terms of the Kronecker invariants of the matrix pencil. Remarkably, this dictionary can be reversed: the invariants of the pencil are read off from intersection numbers and other invariants of the variety. In particular, we prove that the Schubert surface determines the regular component of the pencil, up to the natural action of $\mathrm{GL}_2 \times \mathrm{GL}_n \times\mathrm{GL}_m$. Finally, we prove that the ideal of Schubert surfaces of matrix pencils in the Plücker embedding is generated in degree $2$ and satisfies Green's property $N_p$ in a certain range.

We study finite determinacy and modality in positive characteristic through three complementary constructions: prescribed-jet transversality, Frobenius deformation directions, and a finite flat critical cover. For matrix germs over an arbitrary field, prescribed-jet transversality to the rank stratification shows that finite left-right determinacy is equivalent to finite codimension of the tangent image. Analogous results hold for pure right equivalence, for one-sided actions over infinite fields, and in specified ranges over finite fields. These methods also yield a characterization of finite contact determinacy for positive-dimensional ideals and imply expected-height, Cohen-Macaulayness, reducedness, and normality results for determinantal ideals. For isolated hypersurface singularities over an algebraically closed field of characteristic $p>0$, we introduce a Frobenius algebra whose length and Loewy structure govern the Milnor number, determinacy, and essential corank. Using a finite flat critical cover, we prove that proper modality coincides with right modality. It follows that, in fixed characteristic, bounded right modality gives an ambient-dimension-independent bound on determinacy and, up to nonsingular quadratic suspension, a finite algebraic parametrization by modular families. The same Frobenius methods yield uniform finiteness results for $F$-jumping numbers and show that the $F$-pure threshold is determined at a single Frobenius level.

This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to $CP^2$, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $CP^2$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $CP^2$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_p$-action for an odd prime $p$ on a rational homology $CP^2$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_p$-actions which are not a finite automorphism of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.

We spread out geometrically irreducible perverse sheaves. Over an arithmetic base scheme, such sheaves extend from the generic fiber to a relatively perverse, universally locally acyclic sheaf over a dense open of the base scheme. As an application, we prove the arithmetic Kashiwara conjecture by Esnault and Kerz for geometric traits in equicharacteristic and generalize the decomposition theorem for arithmetic complexes to fields finitely generated over a separably closed field. We also recover the Hard Lefschetz theorem for arithmetic perverse sheaves.

We prove that the resonance of any hyperplane arrangement is separable. Using the main result from arXiv:2512.10160 this proves Suciu's conjecture on the fundamental group of every hyperplane arrangement, in its most effective form.