Papers

We report a proof found by an artificial intelligence system for the following statement: The hyperbolicity cone of every strictly hyperbolic polynomial is spectrahedral. This establishes the generalized Lax conjecture for the generic case, while the full conjecture remains open.

2-step ideals are ideals $I$ of the polynomial ring that satisfy $\mathfrak{m}^{k+2}\subsetneq I\subsetneq \mathfrak{m}^k$ where $\mathfrak{m}$ is the maximal ideal generated by variables. This class of ideals was recently introduced in [F. Giovenzana, L. Giovenzana, M. Graffeo, P. Lella: New components of Hilbert schemes of points and 2-step ideals, 2025, arxiv: 2507.02789] with the aim to obtain new irreducible components of $\mathrm{Hilb}^d(\mathbb{A}^n)$ and in particular to obtain new loci in $\mathrm{Hilb}^d(\mathbb{A}^3)$ of large dimension. In this paper we use the correspondence between Hilbert schemes and varieties of commuting matrices to define loci of commuting matrices that correspond to 2-step ideals. Then we estimate the dimensions of the obtained loci to get new proofs for the estimates of dimensions of loci of 2-step ideals in the case $n=3$. In this case we are also able to omit some assumptions in the dimension estimates in the above mentioned paper.

Let $f\colon X \to Y$ be a proper and tame morphism of algebraic stacks, where $X$ and $Y$ are locally of finite type over an algebraic stack $S$. We prove that $R f_*$ sends complexes that are pseudo-coherent relative to $S$ to pseudo-coherent complexes relative to $S$. In the scheme case, this resolves a conjecture of Illusie from SGA6. We also prove related and new results in the non-tame setting (e.g., infinite stabilizers). Our methods use derived algebraic geometry and also give new proofs of classical statements for schemes due to Kiehl. Along the way, we extend some foundational results for quasi-coherent sheaves on algebraic stacks to the derived setting.

Matrices and tensors are ubiquitous throughout computer vision. The trifocal tensor $\caT$ is a $3\times 3\times 3$ tensor that plays a vital role in three-view geometry. However, this tensor, though displayed in a third-order form, is manipulated and operated primarily in a matrix style in standard literature. This article establishes the conformal tensor algebra by replacing ad-hoc matrix collections with well-defined multilinear operators. We treat all three views symmetrically by covariant subscript indexing ($T_{ijk}$). To make precise what is new beyond notation: the classical point, line, and mixed correspondence constraints are here re-derived, but uniformly, each as a single contraction in one shared algebra; the results that are genuinely new include the exact rank characterization of the trifocal tensor — all mode-$k$ unfoldings $\caT[k]$ in $\RR^{3 \times 9}$ satisfy $\rank(\caT[k])=3$ for non-degenerate setups, and rank deficiency of any unfolding certifies degeneracy of the camera configuration, which yields a near-zero-cost degeneracy alarm and a validation criterion for estimated tensors; the outer-product representation $\caT = A^{\top}\times \bfb_{4} - B^{\top}\times_{2} \bfa_{4}$; and direct epipole extraction via double contractive traces. The coordinate-free multilinear operators seamlessly map to tensor auto-differentiation frameworks (e.g., PyTorch, TensorFlow); a concrete PyTorch experiment demonstrates that the induced $24$-parameter bilinear parametrization acts as a structural prior that eliminates the accuracy drift exhibited by an unstructured $27$-entry autodiff refinement.

We propose a local-cohomology obstruction to the existence of small Cohen–Macaulay modules over completed section rings of surfaces. Let $Y$ be a smooth connected complex projective surface, and let $H$ be an ample globally generated divisor with $K_Y\equiv4H$ numerically. Put $c(Y,H)=15H^2/8-\chi(\OO_Y)$. The main claim is that, when $c(Y,H)>0$, every nonzero finite reflexive module $M$ over the completed vertex local ring of $\bigoplus_{n\geq0}H^0(Y,\OO_Y(nH))$ satisfies $\dim_\C H^2_\mm(M)\geq c(Y,H)\rk M$. The argument combines elementary transformations of reflexive extensions, Harder–Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on $M$. For the degree-six member of the six-line Hirzebruch–Kummer family, $H^2=1080$ and $\chi(\OO_Y)=1926$, so the bound is $99\rk M$. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen–Macaulay modules.

We prove that the solution functor from bounded good filtered complexes of coherent D-modules on a smooth rigid-analytic variety to modules over the positive almost de Rham period sheaf is fully faithful. On filtered vector bundles with flat connection satisfying Griffiths transversality, the construction recovers, up to duality and extension of scalars, Scholze's horizontal sections functor. As an application, we obtain a notion of de Rham Zariski-constructible sheaves, generalising the classical notion of de Rham local systems.

We abstract and prove some fundamental lemmas in universal blowups

Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $\Omega$ be a nonvanishing holomorphic two-form near $F$. We show that a small neighbourhood of $F$ is a Weinstein domain for $\mathrm{Re}\,\Omega$ whose completion is a Legendrian surgery, with cocores obtained by completing holomorphic disks transverse to the components of $F$. For every coefficient field and every multiplicative bulk class, we compute the wrapped Floer cohomology of these cocores and prove that it is concentrated in degree zero. The cocores generate, so the bulk-deformed wrapped Fukaya category is equivalent to the category of perfect modules over an explicit algebra: a multiplicative preprojective algebra of affine type for the normal crossing fibres, and a quiver algebra with relations for types $II$, $III$ and $IV$. Applications include formality of the affine plumbing dg algebras; mirrors given by resolved affine surfaces at every classical bulk class for affine D, E and types II, III, IV; and mirrors given by quotient stacks of algebraic tori at root-of-unity bulk classes for the four affine stars.

In this paper we study the Borel summability and quantum modularity of the free energy computed by refined topological recursion on the Weber-type refined spectral curve. On the one hand, we verify expected properties, namely, Stokes rays come in a pair and the Borel-Laplace sum of the free energy coincides with the Barnes double gamma function. Somewhat unexpectedly, on the other hand, we prove that the generating function of the Stokes constants along each ray is a Jacobi quantum modular function. Our proof suggests that quantum modularity may be a general feature of refined topological recursion, applicable beyond the Weber-type refined spectral curve.

We provide a complex-geometric approach to the Blaschke conjecture, i.e., that a manifold whose injectivity radius equals its diameter is isometric to a compact rank-one symmetric space (CROSS). In particular, we introduce the great quadric fibration, a complex analogue of the great sphere fibration. Requiring its total space to be a complex submanifold simultaneously explains many properties that a Blaschke manifold is expected to possess, including its Clifford structure, Hopf fibration, and diffeomorphism class. Furthermore, the great quadric bundle coincides with the variety of minimal rational tangents (VMRT) of the ambient Fano variety in the standard CROSS cases, and we explain this through a bend-and-break argument in the setting of the Blaschke conjecture under a suitable complexification assumption. Combining this with Hwang-Mok VMRT recognition, we prove the Blaschke conjecture for manifolds admitting an adapted complex structure on their entire tangent bundles.

For a complex $\varepsilon$-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least $\varepsilon^2/100$ and that every integral nef and big Weil divisor has volume at least $\varepsilon^7/1728$. Both exponents are optimal. For $\varepsilon$-log canonical log Calabi–Yau surface pairs of arbitrary Picard number, we prove a lower bound of order $\varepsilon^9/(1+\log_2(1/\varepsilon))$, without a bigness assumption or further restrictions on the boundary coefficients. We also prove $\tau(X,H)\le18\varepsilon^{-3}$ for the pseudo-effective threshold on every projective $\varepsilon$-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order $\varepsilon^{-11/2}\sqrt{1+\log_2(1/\varepsilon)}$ in the log Calabi–Yau setting.

We study the relationship between the integrals of Yuan's concave transforms and the characteristic volumes of adelic line bundles on projective varieties. We establish an identity in dimension one and an inequality in higher dimensions. These results yield counterexamples to an identity conjectured by Yuan.

We prove that every smooth Fano threefold of characteristic p>5 is quasi-F-split. Moreover, if p>11, then it is F-split.

Let $X$ be a smooth minimal projective variety of general type of dimension $n\geq 4$ over $\bb C$. We prove that $|\frac{n^2+n}{2}K_X|$ is base point free. This gives a new bound for the pluricanonical case of Fujita Freeness Conjecture for nef and big canonical divisors.

Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish–Chandra centers of the associated Kac–Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $\kappa$ for $G$, consider the associated Kac–Moody vertex algebra $V_\kappa(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish–Chandra center of arc group invariants $$V_\kappa(\mathfrak{g})^{\mathscr{J}G} \subset V_\kappa(\mathfrak{g}).$$ Write $\check{\kappa}$ for the dual level for $\check{G}$, $\check{\kappa}^p - \check{\kappa}$ for its image under the Artin–Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}$ for the moduli space of $(\check{\kappa}^p - \check{\kappa})$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_\kappa(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}.$$ There is a similar identification for the Harish–Chandra center of the filtered complete enveloping algebra at level $\kappa$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. The presence of these Harish–Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish–Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.

We construct gauge-origami systems on the toric Calabi–Yau fourfold $\mathcal{C}\times\mathbb{C}$ from its brane brick model. Brick matchings and phase boundaries determine elementary D6- and D4-brane framings, respectively, realizing the magnificent-four, tetrahedron-instanton, and spiked-instanton sectors in a unified framed quiver quantum mechanics. We define their partition functions by Jeffrey–Kirwan residues and describe the fixed points in terms of colored BPS crystals. General gauge-origami configurations are obtained by combining these elementary sectors through their common fractional D0-brane quiver. This construction provides a first step toward gauge origami on general toric Calabi–Yau fourfolds.

Let $\pi:\X\to\D$ be a proper holomorphic submersion with connected fibers, and suppose that $X_t$ is biholomorphic to a fixed smooth Moishezon manifold $S$ for every $t\ne0$. If $\kappa(S)\ge0$, we prove that $X_0\simeq S$.

Using N. Honda's construction of twistor spaces, bimeromorphic invarince of $\partial\bar\partial$-lemma for threefolds and embedded resolution, we find a smooth compact complex threefold satisfying the $\partial\bar\partial$-lemma that contains a smooth compact surface for which the lemma fails. This disproves heredity in the smallest possible ambient dimension and yields counterexamples to L. Alessandrini's modification question in every complex dimension at least four. We establish a derived blow-up formula for generalized Bott–Chern complexes with coefficients, compatible with the comparison maps to coefficient cohomology. We also obtain a Dolbeault blow-up formula with bounded derived coefficients. For compact complex manifolds, we introduce a relative $\partial\bar\partial$-property, characterize it using the Frölicher spectral sequence and Hodge filtrations, and prove a blow-up criterion. Finally, counterexamples show the limitations of the classical formulae for coherent coefficients and singular ambient spaces, and the failure of the naive Bott–Chern Künneth formula.

In characteristic $p =3$, we establish a concrete relationship between the torsion rank of the elementary abelian group of global sections of quasi-elliptic surfaces over the projective line ${\mathbf P}^1$ and the $p$-rank of certain hyperelliptic curves, thereby explaining a mysterious relationship between these two invariants previously observed by I. Dolgachev. As an application, we determine the torsion ranks of the Mordell–Weil groups of a class of quasi-elliptic surfaces.

Let $\pi: X\to \mathbb P^1$ be a semistable Jacobian elliptic surface over $\mathbb C$, and set $\chi=\chi(\mathcal O_X)\ge3$, so that $\kappa(X)=1$. Assume that the Mordell-Weil group of $\pi$ is finite and that $\pi$ has at least one reducible fiber, the reducible fibers being of types $I_{n_1},\cdots, I_{n_s}$. Recently, Laface et al. proved that the zero section and the components of the reducible fibers generate $\overline{\mathrm NE}(X)$ if and only if $$\delta(\pi):=\sum_{i=1}^s\frac{\lfloor n_i^2/4\rfloor}{n_i}\le\chi.$$ In particular, the Mori cone is rational polyhedral in this range. They also proved that $N(\pi)=\sum_{i=1}^s n_i\le 2\chi+3$ implies that $X$ is a Mori dream surface. In this paper, we study the existence problem of Mori dream surfaces provided that $N(\pi)\ge 2\chi+4$. Suppose $\delta(\pi)\le \chi$. We first show that every nef isotropic divisor on $X$ is semiample, and whenever each $n_i$ is even. Furthermore, when every $n_i$ is even, we obtain criteria for $X$ to be a Mori dream surface: $X$ is a Mori dream surface provided that either (i) all $n_i=2$, or (ii) $N(\pi)\le 2\chi+4$; if some $n_j=4$, then $X$ is a Mori dream surface if and only if $N(\pi)\le 2\chi+4$. However, once some $n_i$ is odd, we construct a Jacobian elliptic surface $\pi: Y\to\mathbb P^1$ with $\delta(\pi)=\chi=3$, trivial Mordell-Weil group, and singular-fiber configuration $I_4+3I_3+23I_1$ for which none of the eight non-vertical isotropic extremal rays is semiample. In particular, $Y$ is not a Mori dream surface.

Let $\Delta$ be a reflexive four-polytope with primitive edges. We construct deformations of the toric fourfold defined by its face fan that smooth the singularities of a generic anticanonical Calabi-Yau hypersurface over a chosen two-dimensional face. The construction applies when the face cone is smoothable and the dual edge has an interior lattice point; its equations are trinomials in Cox coordinates. It gives a smoothing of the hypersurface when this is the only face whose cone is singular. If every singular face satisfies these conditions, simultaneous smoothing follows in the nodal case and, in general, under irreducibility of the semiuniversal deformation space. A seven-vertex example with one ordinary double point and no smoothing shows that the dual-edge condition is sharp. A scan of the Kreuzer–Skarke classification shows that the facewise criterion accounts for 3,774 of the 30,241 smoothings in the Batyrev–Kreuzer census; the other 26,467 require relations among exceptional curve classes from different faces.

The Fresnel wave surface governs the propagation of light in a transparent biaxial crystal. It is a special Kummer quartic, and we identify it exactly. Over the complex numbers the wave surface of a crystal is the Kummer surface of the Jacobian of an explicit genus-two curve branched at the signed square roots of the three principal permittivities. This Jacobian is isogenous, by an isogeny with kernel of order four, to a product of two elliptic curves. One elliptic curve carries the three permittivities, and the other carries the optic-axis angle. The physical family is Zariski dense in the locus of genus-two curves with an extra involution, and its automorphism strata are explicit. The identification instantiates a task-aware quotient, which identifies parameters that differ by a nuisance transformation and carries invariant coordinates and explicit strata. For biaxial crystals, two ratios of the permittivities form a complete invariant of the wave surface up to rotation and rescaling, and the four real nodes are given in closed form. At an interface the candidate transmitted waves are the roots of a quartic of exact degree four. Its real-root count, root order, and repeated-root events are decided by exact algebraic predicates, and along the generic single-node encounters of the paper its discriminant vanishes to second order. Floating-point solvers drop forward transmitted modes near the optic axes, and the certified solver does not. On exact equivalence classes, learned models on quotient coordinates are invariant and more accurate than models on raw tensors, while learned root-count predicates fail near the optic axes.

We relate the monodromy group of a branched cover to the monodromy groups of its restrictions. We show that the presence of a transposition in a restricted monodromy group significantly constrains the monodromy group of the original branched cover. Using this result, we complete the sparse trace test algorithm, which numerically verifies whether a set of solutions to a sparse polynomial system is complete. Along the way, we develop a generalization of the notion of a trace test in the setting of branched covers.

The Albanese fibration of the Cartwright–Steger surface is a genus-$19$ Lefschetz fibration over a torus. We give an explicit formula for it: in the projective model of Borisov and Yeung, the map is represented by three homogeneous polynomials of degree $13$ with integer coefficients, and its target is the elliptic curve $Y^2Z=X^3-3888Z^3$. A recognition theorem shows that it suffices to verify one polynomial identity on the surface. A coefficient bound allows us to establish the identity over $Q$ from computations modulo primes. We also determine exact coordinates for the three nodes and their images on the elliptic curve.

Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $\pi:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $\pi$. For $\sigma\in S_d$, write $\operatorname{ind}(\sigma)$ for its permutation index, namely $\operatorname{ind}(\sigma) = d-\#\{\text{orbits of $\sigma$ on $\{1,\ldots,d\}$}\}$. If $\tau$ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most $X$ is bounded below by a constant multiple of $X^{1/(2n\,\operatorname{ind}(\pi(\tau)))}$. For the natural vector and projective actions, the exponents we obtain are asymptotically $1/(4n)$ of those predicted by the weak form of Malle's conjecture as $\ell\to\infty$. The fields are constructed from the mod-$\ell$ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.

Exact calculations for smooth toric Fano varieties in dimensions at most seven suggest a sharp dimension-dependent lower bound for the positive difference between the rational and integral Gongyo indices. We compute both indices for the smooth toric Fano models obtained from blow-ups of projective space at torus-invariant points by the anticanonical minimal model program. For pseudo-symmetric smooth toric Fano varieties, we prove the proposed lower bound and determine all equality cases. We also apply a dimension-raising construction to obtain an explicit infinite family of smooth toric Fano varieties with positive gap from odd-dimensional weak Fano varieties with zero gap. For the standard models considered, we determine the smallest dimension increase needed to obtain a Fano variety by iterating this construction and compute both indices of the minimal lifts. Finally, every positive rational number occurs as a Gongyo-index gap, even within the pseudo-symmetric class.

We consider local cohomology sheaves on compact Hermitian symmetric spaces supported in Schubert varieties, which carry natural structures as mixed Hodge modules. Our main result gives an explicit root-theoretic construction, uniform across Lie types, for the D-simple composition factors and weight filtration on these modules. For Lagrangian and orthogonal Grassmannians, we translate these formulas into combinatorial rules involving strict partitions and shifted Dyck patterns, analogous to existing formulas on ordinary Grassmannians. As an application, we expand upon work of Raicu–Weyman to describe the weight filtration on local cohomology supported in symmetric determinantal and Pfaffian varieties. We also give formulas for the Hodge rational homology level and local cohomological defect of every Schubert variety in a compact Hermitian symmetric space. Finally, we tabulate the local cohomology weight filtrations for all Schubert varieties in the two exceptional Hermitian pairs.

We prove Steinness over $\breve{\mathbf{Q}}_p$ for the basic local Shimura tower attached to $G=\mathrm{GL}_n$, $\mu=(1,1,0^{n-2})$, with basic Newton slope $2/n$, for every odd $n\ge3$, at every finite level and for every prime $p$. The proof proceeds by constructing global analytic functions from the crystalline period map, proving compactness of their sublevel sets via a perfectoid normalization of Tate lattices, and showing that the resulting function map to affine space is finite. The key geometric inputs are the Fargues–Fontaine realization of the universal cover, a normalized determinant on the rank-two locus, and an integral PEL realization over $W(\overline{\mathbf{F}}_p)$ used to produce bounded global Hodge generators. The integral constructions include the prime $p=2$.

We study a natural class of normal functions on the moduli space of pointed curves and determine their rank at every point. We prove that the rank is maximal outside the corresponding stratum of Abelian differentials, while along this stratum it drops by exactly one. As a consequence, we describe the local geometry of the associated double ramification locus.

We classify compact Kähler–Einstein surfaces whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space. No symmetry assumption and no bound on the codimension are imposed. We prove that the only such surfaces are \[ (\PP^2,m g_{\FS}) \quad\text{and}\quad \bigl(\PP^1\times\PP^1,m(g_{\FS}\oplus g_{\FS})\bigr), \qquad m\in\mathbb Z_{>0}, \] realized respectively by the Veronese and Segre–Veronese embeddings. The main new ingredient is a codimension-independent exclusion of the entire Fano-index-one branch, combining a common anticanonical root construction with Gram–Gauss rank estimates and, in degree five, an equivariant curvature argument. Consequently, every connected compact Kähler–Einstein surface whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space is homogeneous.