Papers

Let $X$ be a planar vector field with an equilibrium $p$ at which the Jacobian $J=DX(p)$ is nilpotent of rank one, and let $q_0$ span $\ker J$. We prove that the two coefficients $a$ and $b$ of the Bogdanov–Takens (BT) normal form are the directional derivatives, along $q_0$, of the two invariants of the Jacobian: $a=-\frac{1}{2}\langle\nabla\det DX(p),q_0\rangle$, $b=\langle\nabla\operatorname{tr}DX(p),q_0\rangle$. The identity is invariant under changes of phase-space coordinates and equivariant under the rescaling of $q_0$, and the resulting formula requires neither generalized eigenvectors nor the second-order multilinear form. It yields a coordinate-free reading of the BT nondegeneracy conditions in terms of the kernel line field of the projection of the equilibrium manifold onto parameter space, the transformation rule $(a,b)\mapsto(h^2a,hb)$ under orbital equivalence, and the fact that $a=0$ whenever the vector field factors through a function vanishing at $p$. We then prove an obstruction of a combinatorial nature. For a Kolmogorov system $\dot{x}=xA/g_1$, $\dot{y}=yB/g_2$ and an equilibrium $p$ in the torus $(\mathbb{C}^*)^2$ at which the Jacobian is nilpotent and nonzero, the order $m=\operatorname{ord}f$ in the Takens normal form is bounded by the mixed volume of the Newton polytopes of $A$ and $B$. In particular, if $\operatorname{MV}(\operatorname{Newt}A,\operatorname{Newt}B)\leq2$ then $a\neq0$ unless the equilibrium fails to be isolated, and the nilpotent singularities of saddle, focus and elliptic type are unreachable: only cusp-type singularities occur at isolated equilibria, while their exact codimension is not controlled by the mixed-volume bound. The class of systems with cross-product cubic terms, for which the mixed volume equals $2$, is treated in detail, and two classical Bazykin models are shown to be instances.

We classify rank $2$ weak Fano bundles on a four-dimensional smooth quadric hypersurface $Q^4$. Up to twisting with a line bundle, such a bundle is either a split bundle, a spinor bundle, or one of the stable bundles with Chern classes $c_1=-1$ and $c_2=(1,1)$ constructed in [APW94].

We count the twisted forms, over the field of real numbers and over finite fields, of the three classical families of smooth hypersurfaces with large automorphism group: the Fermat, Delsarte and Klein hypersurfaces. Our main tool is a counting formula for the Galois cohomology set of a smooth hypersurface whose automorphism group is the semidirect product of a diagonal abelian group and a group of permutations of the monomials of its defining equation. In order to apply this formula over finite fields, we extend the differential method of Oguiso and Yu to positive characteristic, and we compute the automorphism groups of the Fermat, Delsarte and Klein hypersurfaces over algebraically closed fields of positive characteristic under explicit arithmetic conditions on p. Over the reals, our count recovers a recent theorem of Sasaki on the real forms of Fermat hypersurfaces.

We give a machine-checked proof, in Lean 4, of the universal property of multicentered blowups: given a finite family $Z$ of closed subschemes of a scheme $X$, presented on each chart by ideals, there is a scheme $\Bl_Z X$ over $X$, terminal among $X$-schemes on which every member of the family becomes an effective Cartier divisor. To the best of our knowledge this is the first formalisation of blowups, single- or multicentered, in any theorem prover. The proof is organised around the universal property of the multicentered dilatation of a ring: the initial algebra in which a finite family of ideals becomes generated by non-zero-divisors. Existence and uniqueness of the dilatation are established locally, on affine charts, and every subsequent step of the construction, the base change along open immersions, the comparison isomorphisms on overlaps, the triple overlap maps, the cocycle identity, and the independence of the chosen presentation, is deduced from that single local property by uniqueness alone. We give the exact theorem statement as formalized, the proof architecture, and the main new mathematical ingredients, together with the corresponding Lean code.

We prove that the canonical irregular Hodge filtrations in smooth families of Deligne–Mumford Landau–Ginzburg models with fixed pole orders assemble into filtered algebraic vector bundles. More precisely, twisted de Rham cohomology forms a vector bundle carrying a canonical algebraic integrable connection, every rational irregular Hodge level is a subbundle with locally free quotient, and every rational graded piece is a vector bundle. These constructions commute with arbitrary finite-type base change, and the connection satisfies shifted Griffiths transversality. Consequently, all filtered and graded dimensions are locally constant. As an application, for a smooth quasiprojective toric Deligne–Mumford stack and every fixed Newton polytope at infinity satisfying the support condition, these conclusions hold over the full coefficient locus of Laurent polynomials that are nondegenerate at infinity. In particular, their filtered and graded dimensions are independent of the coefficients. When the resulting stacky fans form a Clarke dual pair, the irregular Hodge numbers are computed by Harder–Lee's combinatorial $\Xi$-complex.

A Lie algebra is said to be $1.5$-generated if every nonzero element can be completed to a two-element generating set. We prove that the Lie algebra of polynomial vector fields with constant divergence on the affine space is $1.5$-generated. We also show that the Lie algebra of all polynomial vector fields on the complex affine space is generated by two completely integrable elements.

Let $U$ be a smooth complex affine log Calabi–Yau surface, and let $\widehat{U}$ be its complete finite-type Liouville manifold, equipped with its natural grading structure. We give an algorithm that constructs a finite-type quasi-projective $\mathbb{Z}$-scheme $U^\vee$ such that \[ D^\pi\bigl(\mathcal{W}(\widehat{U},\Bbbk)\bigr) \cong D^b\operatorname{Coh}(U^\vee_{\Bbbk}) \] for every field $\Bbbk$. Our main contribution is to construct an almost toric model encoded by an exact eigenray diagram and, using the symplectic Torelli theorem for symplectic log Calabi–Yau pairs, to prove that this model is grading-preserving strongly exact symplectomorphic to $\widehat{U}$. The result then follows from the homological mirror symmetry theorem of Hacking–Keating.

Let $\mathbb{K}$ denote an algebraically closed field of arbitrary characteristic. In this paper we provide bounds for the size of a prime $\ell$ for which there are curves defined over $\mathbb{K}$ admitting automorphism groups of order $\ell^2$. In addition, we also give a classification of the families of curves attaining the upper bounds for $\ell$ in both tame and wild case. Finally, we present the full automorphism groups of the curves attaining the highest bounds.

Let $y^2 = f(x,T)$ be a hyperelliptic curve of genus $g\geq 1$, defined over $\mathbb{Q}(T)$. We prove the existence of infinitely many imaginary hyperelliptic curves with a fixed genus $g$ having a certain rank for $5\leq r\leq 4g+2$, and a similar result for real hyperelliptic curves with a fixed genus $g$ having a certain rank for $6\leq r\leq 4g+4$. We begin by constructing such curves and prove the rank using two methods. First, we apply the generalized Nagao's conjecture, which relates the first moment and the rank of the Jacobian variety $J_\mathcal{X}(\mathbb{Q}(T))$, and that the conjecture holds for our curves, making the result unconditional. Furthermore, we explicitly construct rational points in the Mordell-Weil group and use Shioda-Tate to prove that the rank is equal to $r$.

We give the minimum of category theory necessary for understanding the definition and applications of Grothendieck topos. We state the basic properties of sites and sheaves, and give applications to the theory of moduli. We prove that for categories $\mathbf C$ with explicit stated properties, we can construct a category of schemes of objects in $\mathbf C$ which is a proper category rather than a $2$-category as in the case of stacks.

Following ideas of Kapustka-Kapustka, we use Lagrangian fibrations to construct twisted hyperholomorphic sheaves on products of hyperkähler manifolds of K3$^{[n]}$- and OG10-type. As applications, we prove the Lefschetz standard conjecture and the D-equivalence conjecture for hyperkähler manifolds of OG10-type.

In this note, we construct moduli spaces of super-representations of quivers by extending King's Geometric Invariant Theory (GIT) framework to the super setting. Using the even part of super general linear groups and a parity-shifting operator, we construct moduli spaces that topologically parameterize super-representations of a fixed super-dimension. We also outline the construction of super quiver varieties for super-representations of framed quivers (without doubling); and realise super-Grassmannians and super-flag varieties as geometric quotients within this setting.

Let $f$ be a non-CM newform of weight $k\geq4$, level $N$ and trivial Nebentypus. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and denote by $\boldsymbol f^{(p)}$ the $p$-adic Hida family passing through $f$. Assuming very specific instances of two general conjectures in arithmetic algebraic geometry (injectivity of $p$-adic Abel-Jacobi maps, positive definiteness of archimedean height pairings à la Gillet-Soulé), we prove that, for all but finitely many $p$ as above, if the analytic rank of $f$ is $1$, then all but finitely many specializations of $\boldsymbol f^{(p)}$ of even weight and trivial Nebentypus have analytic rank $1$. This result provides evidence for Greenberg's "minimality conjecture" on analytic ranks in families of modular forms.

We tropically enumerate planar well-spaced elliptic curves with a fixed $j$-invariant. Combined with the recent genus 1 correspondence theorem of Cela–Koyama, this yields an alternative proof of Pandharipande's algebraic enumeration.

Let $\mathfrak g$ be a complex simple Lie algebra of type $G_2$, $F_4$, or $E_8$. We prove that the closure $\overline{\mathcal O}_{\min}(\mathfrak g)$ of its minimal nilpotent orbit is not isomorphic to $(T^*X)^{\mathrm{aff}}$ for any smooth quasi-affine variety $X$. We do not assume that the isomorphism is equivariant or Poisson. We also prove the same statement in types $A_1$ and $A_2$.

Let $Y$ be a very general prime Fano threefold of genus 7. We exhibit an explicit 2-cycle on $Y\times Y$ that is Abel-Jacobi trivial but non-torsion in the Chow group $A^4(Y\times Y)$. As a consequence, $Y$ does not admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We also show that any Fano threefold has a multiplicative Chow-Künneth decomposition modulo algebraic equivalence.

In 1972, M. Sato proposed an approach to proving modularity of forms like Thetanullwerte by characterizing them via certain differential operators of infinite order (DOI) in the modular variable(s) alone. A DOI is an infinite series in derivatives decreasing so fast that it acts on holomorphic functions by a sheaf morphism. This approach was developed by several authors including Kashiwara, Kawai, Takei and Yoshida. We give an interpretation of this approach using supersymmetry which provides a natural source of DOIs: the naive exponential of any odd supersymmetry generator is a DOI. The case of the Riemann theta function of genus n is governed by the supergroup OSp(1|2n) (and its metaplectic cover) acting on a natural super-thickening of the Siegel plane. For n=2 this is the 3-dimensional N=1 superconformal group and the structure at hand is precisely the free massless scalar supermultiplet (combining the Laplace and Dirac equations). For n>2 we get a super-extension of the generalized conformal structure existing on the Lagrangian Grassmannian as on any Hermitian symmetric space. An additional interesting feature here is that the odd supersymmetry generators acting “on-shell” (i.e., in the space of solutions of the equations of motion) satisfy even-style Heisenberg commutation relations. These equations of motion upgrade to a complex of differential operators corresponding to a natural BGG-type resolution of the super-Weil representation of osp(1|2n).

We study whether the topology underlying a Calabi-Yau manifold can support natural geometric positivity or negativity structures. In even complex dimension $n \geq 4$ (assuming $b_2=1$ when $n \geq 6$), we strengthen a theorem of Oguiso-Peternell by proving that a Calabi-Yau manifold is not homeomorphic to a weak Fano $n$-fold. The same obstruction applies to Kähler manifolds with quasi-positive holomorphic sectional curvature. A transformation-group analogue excludes, in particular, symplectic manifolds admitting Hamiltonian circle actions with isolated fixed points. On the negative side, we show that the fundamental group of a Calabi-Yau manifold is not isomorphic to that of a Kähler hyperbolic manifold. Taken together, these results exhibit a common topological rigidity separating Calabi-Yau manifolds from several fundamental classes governed by geometric positivity or negativity.

We establish lower bounds for the Smith–Thom deficiency of the Hilbert square of a maximal nonsingular real projective variety. These bounds recover the known surface case and provide new obstructions to Smith-Thom maximality in every dimension at least two. As applications, we prove that the Hilbert square of any real abelian variety of dimension at least two is not Smith-Thom maximal. We further show that the deficiencies of the Hilbert squares of maximal real abelian varieties grow exponentially with the dimension. We also obtain nonmaximality results for the Hilbert squares of certain Cartesian products.

We introduce the tame étale fundamental group $\pi_1^t(X/K)$ of a rigid space X over a non-archimedean field K. We show that if X is qcqs and K has topologically finitely generated tame Galois group (e.g. algebraically closed or a local field), then $\pi_1^t(X/K)$ is topologically finitely generated. If X is moreover the rigid generic fibre of a strictly semistable formal scheme such that the smooth locus of its special fibre admits a projective snc compactification, then $\pi_1^t(X/K)$ is topologically finitely presented. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids), in particular on an analogous finiteness statement for the tame log étale fundamental group, and on the 'vertical compactification' of a map of adic spaces.

Let $A$ be a finitely generated commutative algebra over a field $\mathbb K$ of characteristic zero. We prove that every finitely generated Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally finite on $A$ is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on $A$, and all its elements are locally finite. A key ingredient is a second theorem: every Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally nilpotent is solvable; when $A$ is reduced, its derived length is at most $\dim A$, and this bound is sharp. In general, the $(\dim A)$-th term of its derived series is nilpotent, of class bounded in terms of $A$ only. We also prove that every solvable Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ generated by finitely many locally finite derivations is finite-dimensional and acts locally finitely on $A$. For an affine variety $X$ over an algebraically closed field, we deduce that a subgroup of $\operatorname{Aut}(X)$ generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this gives an answer to a 2005 problem of Popov. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proof uses derivations vanishing to second order along a suitable finite subscheme, on which local finiteness forces local nilpotence.

We associate to a collision $p\ne q$ of a homogeneous Keller map $F=\operatorname{id}+h$ the polarization subalgebra generated by $p$ and $q$. This collision hull is the smallest invariant linear subspace containing the collision; the restricted map remains a noninjective Keller map, and its dimension controls that of the standard symmetric lift. The construction is compatible with scalar extension and equivariant under linear conjugacy, so it gives a canonical linear carrier for a marked failure of injectivity. We compute this carrier exactly in two recent cubic-homogeneous reductions of the three-variable Jacobian counterexample. For Thompson's 24-variable map, the growth $2,4,11,20,20$ recovers MacFarlane's 20-dimensional invariant subspace. For the 19-variable homogenization of Van Rijn's 12-variable degree-three map, the growth $2,4,11,19,19$ fills the whole space. Hence no invariant linear restriction retaining the displayed collision can improve the corresponding 38-variable symmetric lift. We give the resulting 340-monomial homogeneous quartic explicitly: it is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over $\mathbb{Q}(i)$. The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.

Automated-driving rulebooks rank rule violations lexicographically, and model predictive control enforces that ranking either exactly, through $L{+}1$ sequential programs per tick, or approximately, through a weighted sum tuned by the separation heuristic $w_1\gg w_2\gg\cdots\gg w_L$. We show the heuristic answers the wrong question. For a convex priority-ordered program, a weighted sum reproduces the lexicographic optimum precisely when its weight, augmented by a unit performance coefficient, supports the upper image of the achievement map at the lexicographic point; the admissible weights form the unit-performance slice of an outward normal cone. Under hinge penalties this slice is a polyhedron obtained by projecting a scaled-KKT system, and a linear program returns an interior weight with a certified margin; under squared-hinge penalties no finite weight is exact whenever the limiting multiplier is nonzero, with violation along the local minimizer branch decaying as $O(1/w)$. Calibrated on held-out logs, the resulting weights have near-equal tier components in nine of eleven calibration-eligible scenario classes and roughly double legal-tier event precision against a matched heuristic weight in closed-loop nuPlan experiments on a 25-rule rulebook. The certificate is, however, pointwise: no single weight is valid across the sampled ticks of an episode, the median lifetime is one sampling interval (zero subsequent ticks at the native rate), and persistence tracks active-set stability. These findings motivate monitored weighted solves with selective cascade fallback, although the compliance-pattern monitor detects only a subset of measured lapses.

We study descendant integrals on moduli stacks of simplicially stable curves. For each finite collision complex, we factor the reduction morphism into elementary wall crossings and obtain a reconstruction formula in terms of ordinary Witten-Kontsevich correlators. We use the formula to derive corrected Virasoro recursions. We then package all finite-support theories in a completed disjoint-support Fock space, where the extended potential is a cumulant translate of the Witten-Kontsevich potential.

The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.

We construct quintic surfaces in the three-dimensional projective space $\mathbb P^3$ with $18$ ordinary cusps. Our starting point is the Barth–Rams description of quintics containing a $3$-divisible set of $12$ cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with $16$ cusps, and examples with $18$ cusps can be found over small finite fields. Our main construction is based on quintics admitting two Barth–Rams decompositions. The corresponding sets of $12$ cusps meet in $7$ points, and we prove that the locus of quintics admitting two such decompositions contains a $6$-dimensional component in the moduli space whose general member has $17$ cusps. This makes it possible to find members with $18$ cusps efficiently over finite fields. We lift one of these surfaces to characteristic zero using Newton–Hensel lifting and LLL reconstruction, obtaining a quintic over a number field of degree $22$. We verify that this surface has $18$ ordinary cusps and no other singularities.

We compute explicit defining equations for eight fake quadrics arising as $\mathbb Z/2\times\mathbb Z/4$-covers of two singular $\mathbb Z/2$-Godeaux surfaces obtained in earlier work of the second author. Starting from explicit equations for the universal covers of the Godeaux surfaces, we reconstruct the relevant character eigenspaces and determine the homogeneous ideals of the bicanonical models of the eight fake quadrics in $\mathbb P^8$. All eight models are defined over $\mathbb Q$. We prove that the surfaces are pairwise non-isomorphic and rigid. Combined with the non-product result established in the earlier work, this gives the first explicit projective models of fake quadrics which are not isogenous to a product of curves.

Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7 + \beta x^6y$ for $\alpha^2+\beta^2 \neq 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg} \, p =7$ and $\mathrm{deg}\, q$ is even and coprime with 7.

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and let $I$ be an ideal of $R$. The epsilon multiplicity $\epsilon(I)$ of $I$ is defined by Ulrich and Validashti in \cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, $$ \epsilon(I)= \lim_{n\rightarrow\infty} \frac{\ell_R((I^n)^{\rm sat}/I^n)}{n^d/d!}. $$ We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity $a\epsilon(I)$. We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.

The compatibility of the semiorthogonal decomposition of a cubic fourfold with Bridgeland stability beyond a generic point is the missing ingredient in the conjectured dynamical protection of the K3 atom $\AX$. We analyze it in the exactly solvable models of the noncommutative minimal model program. In the uncoupled Fano models ($\PP^1$; $\PP^1 \times \PP^1$ at resonance) the quantum cohomology path exits the geometric chamber at an explicit finite time, enters the selection region of the gluing, and never leaves; perturbing the resonance shows the $\varepsilon$-crossover is not a wall. The $A_2$ quiver provides a coupled counterpart. We examine the deformed cubic oscillator, Painlevé~I, and formulate a tau-JLO dictionary as a determinant identification and establish its perturbative layer, including computing closed-form quantum periods through $Z_4$, all-orders flatness with exact curvature the tau-divisor current, identifying it with the zeta determinant's zero divisor along a Painlevé~I trajectory. We close by conjecturing that the determinant's nonperturbative jumps are automorphisms of the underlying Joyce structure.