Daily Math Paper Digest
Miscellaneous topics in algebraic geometry.
Xu and Zhuang prove that the CM line bundle is ample on every proper K-moduli subspace whose geometric points represent reduced uniformly K-stable Fano varieties. Their algebraic proof turns stability into positivity for Harder--Narasimhan filtrations, realizes torus corrections by twisted families, and obtains uniform positivity along movable curves.
Rams and Schütt prove that a polarized K3 surface of degree (2h), in characteristic different from (2) and (3), has at most 24 rational curves of degree at most (d) once (h>42d^2), and they construct sharp families. The proof turns the curves' intersection graph into a lattice problem and forces any extremal configuration into a genus-one fibration.
Frei proves that the rational ell-adic Galois representations of a smooth proper moduli space of stable sheaves on a K3 surface are determined by those of the surface together with the dimension. Over a finite field, equal zeta functions for two K3 surfaces therefore force equal zeta functions for equal-dimensional sheaf moduli spaces, even without birationality.
Tayou proves that a non-isotrivial elliptic K3 surface over an algebraically closed field of any characteristic has infinitely many rational curves, and obtains the same conclusion for every elliptic K3 surface outside characteristics two and three. The proof combines divisible Brauer classes, rational multisections, torsion monodromy, and an isotrivial fiber-classification argument.
Minoccheri and Pan use bend-and-break and the geometry of conic evaluation fibers to prove vanishing of first Griffiths groups and line generation of first Chow groups for broad classes of Fano and 2-Fano varieties.
Barros proves that the moduli space of genus-11 primitively polarized K3 surfaces with n ordered marked points is unirational for n at most 6 and uniruled for n at most 7. The argument relates pointed K3 surfaces to normalizations of nodal hyperplane sections and then to pointed curve moduli.
In the singular rank-at-least-two regime, the degree-zero Higgs-bundle moduli space is a symplectic singularity and has a projective symplectic resolution exactly for an elliptic curve or in genus two and rank two. Formal isosingularity transfers local information from character varieties, while symmetric products give the elliptic resolutions.
Diverio, Fontanari, and Martinelli prove that a smooth projective complex manifold of dimension greater than two with finite fundamental group contains a rational curve whenever it admits a one-dimensional fibration and its canonical bundle is pulled back from the base. The proof forces a divisorial discriminant, slices to an elliptic surface, excludes multiple fibers by global generation, and applies Kodaira's classification of singular fibers.
Fujita and Odaka introduce the delta invariant from log canonical thresholds of basis-type anticanonical divisors and prove that delta greater than one implies uniform K-stability, while delta at least one implies K-semistability. Their proof converts basis filtrations into divisorial volume integrals and then applies a valuative criterion for K-stability.
Near a singular point of a moduli space of pure one-dimensional sheaves on a K3 surface, the authors identify the analytic germ with a Nakajima quiver variety. Changing the polarization to a neighboring chamber matches variation of the quiver GIT character, including the resulting symplectic resolutions.
For rank-two parabolic bundles on the projective line, the authors compute the effective cone at general maximal-Picard weights and identify its extremal rays with level-one conformal blocks. They then show that every interior Mori model is again a moduli space of parabolic bundles.
Codogni, Fanelli, Svaldi, and Tasin give monodromy criteria for deciding when a terminal Q-factorial Fano variety occurs as the general fibre of a Mori fibre space. For rigid Fano varieties, the criterion becomes an exact characterization in terms of automorphism-invariant divisor classes.
Tosatti characterizes finite Weil-Petersson distance for a degeneration of smooth Calabi-Yau manifolds by the existence, after finite base change, of a birational model with canonical Calabi-Yau central fiber. The proof joins an L2 criterion for canonical singularities to semistable reduction and the relative minimal model program.
Gagliardi and Hofscheier classify Gorenstein Fano embeddings of a fixed spherical homogeneous space by reflexive polytopes modified by colors and the valuation cone. Their polyhedral description yields the sharp Picard-number bound rho(X) <= 2 dim(X), with equality only for products of the degree-six del Pezzo surface.
Addington and Thomas compare Hassett's Hodge-theoretic K3 condition for a cubic fourfold with Kuznetsov's categorical condition. They prove that the K3 category is geometric on a nonempty Zariski-open dense subset of every admissible Hassett divisor, then use algebraic cycles to extend the associated Hodge isometry over the whole divisor.
Bayer and Macrì attach a canonical nef divisor to any proper family of Bridgeland-semistable objects, with zero degree exactly along curves of S-equivalent objects. For K3 surfaces this divisor proves projectivity and ampleness in generic chambers and turns suitable stability walls into birational contractions or flops.
Kuznetsov constructs faithful base change for semiorthogonal decompositions of bounded derived categories. The proof passes through perfect and unbounded quasicoherent complexes, then uses an ampleness criterion and finite cohomological amplitude to recover bounded coherent components after base change.
Wang characterizes algebraizable topological real vector bundles on smooth affine real varieties of dimension at most three using the first two Stiefel--Whitney classes. In dimension four, a motivic Postnikov obstruction involving Pontryagin and higher Stiefel--Whitney data appears.
Friedman proves unobstructedness for compact singular Calabi-Yau varieties with isolated local-complete-intersection Du Bois singularities, vanishing first structure-sheaf cohomology, and a resolution satisfying the partial-partialbar lemma. The proof converts relative Hodge-theoretic surjectivity into the T1-lifting property and also treats non-lci, log Calabi-Yau, and weak Fano variants.
Ottem and Rennemo construct smooth Picard-rank-one Fano varieties in every even dimension at least four whose third integral cohomology is Z/2. In dimensions at least six, the torsion generator also separates coniveau from strong coniveau through a nonzero mod-2 square.
Relative log-canonical singularities of an algebraic stack force the stabilizer at the point to be a finite extension of a torus and yield an étale-local good moduli space; relative canonical singularities additionally force a nonempty stable locus. The proof converts the presentation foliation into a semistable relative tangent foliation and uses its faithful semisimple normal representation to exclude additive stabilizers.
Fry constructs modular compactifications of rational marked curves from a graph and proves that their boundary cone complex is realized faithfully by geometric tropicalization exactly for complete multipartite graphs. Plucker coordinates, divisorial valuations, cross-ratio units, and a forbidden three-vertex pattern identify the precise combinatorial obstruction.
The good moduli spaces of semistable rank-r vector bundles on a smooth projective curve of genus at least two, with either fixed degree or fixed determinant, are projective varieties. The proof combines modern existence criteria for proper good moduli spaces with descended determinantal line bundles whose cohomology sections are shown to be semiample and point-separating by dimension counts and elementary transformations.
Every complex projective elliptic K3 surface contains rational curves of unbounded self-intersection, and the lifts of its rational curves are Zariski-dense in the projectivized cotangent bundle. The proof uses non-quasi-torsion multisections and fiberwise multiplication to manufacture singularities and distinct tangent directions.
Kara, Portakal, and Tsuchiya characterize when a directed edge polytope has only triangular two-faces, hence when the associated terminal Gorenstein toric Fano variety satisfies Totaro's rigidity criterion. The characterization is expressed by two explicit forbidden or required local configurations in the directed graph.
For every smooth Fano threefold, the number of irreducible-domain rational-curve components of bounded anticanonical degree grows at most polynomially. The proof combines a classification of non-dominant families with a movable bend-and-break theorem for free curves.
Zhan proves that monodromy permutes the 24, 324, and 3200 rational curves in the relevant low-genus K3 linear systems as the full symmetric groups. The genus-three proof combines incidence-space irreducibility with a specially degenerated quartic whose single cuspidal rational section produces a transposition.
Loginov proves that the dual complex of a semistable Fano degeneration is always a simplex and classifies the maximal degeneration in dimensions at most three. Connectedness for log Fano strata gives the simplex, while low-dimensional MMP and d-semistability force the unique flag blow-up model.
Restriction from a Fano threefold to an anticanonical K3 surface produces Lagrangian subvarieties in moduli spaces of bundles when an obstruction group vanishes. Serre's construction turns curve-theoretic vanishings into many concrete examples, including Lagrangians in the O'Grady tenfold.
Araujo, Fassarella, Kaur, and Massarenti determine the full automorphism group of the central-weight moduli space of rank-two parabolic bundles on a marked projective line. They show that every automorphism is an elementary transformation by reconstructing the parabolic ruled surface from the Hitchin system and its nilpotent cone.
Stibitz and Zhuang connect birational superrigidity to K-stability: a Picard-rank-one Q-Fano variety with log-canonical movable anticanonical boundaries and alpha invariant at least one half is K-semistable, with stability under the strict threshold and, in the superrigid case, even at equality. The proof converts the valuative beta criterion into a sharp restricted-volume inequality.
Totaro constructs smooth Fano varieties in every positive characteristic on which Kodaira vanishing fails, turns half-anticanonical examples into terminal non-Cohen--Macaulay cones, and gives an explicit terminal non-Cohen--Macaulay threefold quotient in characteristic two.
Zhuang proves in positive characteristic that a log Fano variety whose anticanonical Seshadri constant exceeds its dimension is projective space, and that the threshold case is globally F-regular when the characteristic is greater than two. The paper also classifies the equality cases and proves boundedness results for threefolds in characteristic greater than five.
For every general linear section of the Plucker Grassmannian Gr(2, 5) of dimension two through six, the paper proves that the moduli spaces of smooth rational curves of degree at most three are rational. It also constructs birational blow-up/down models showing that the conic Hilbert schemes on the four- and five-dimensional sections are smooth and irreducible.
The anti-canonical volume of a Kähler–Einstein Q-Fano variety is bounded by local singularity invariants, either log-canonical-threshold times multiplicity or normalized valuation volume. A blow-up volume estimate and Ding-semistability turn local vanishing into a global bound, yielding sharp quotient-singularity estimates and a cone criterion for K-semistability.
The Fermat quartic K3 surface has the Hilbert property over the rationals despite being non-unirational, proved by combining two elliptic fibrations with ramification and thin-set arguments. The paper also shows that the Hilbert property forces algebraic simple connectedness and constructs a related Enriques surface with dense rational points but without the Hilbert property.
The CM line bundle on the proper moduli space of smoothable K-polystable Fano varieties carries a canonical continuous metric whose positive curvature current extends the Weil-Petersson form. Its positivity makes the CM bundle nef and big and yields a projective embedding of the smooth Kähler-Einstein locus, proving that locus is quasi-projective.
The moduli space of semistable bundles of rank r and degree d on a real elliptic curve is identified, over the real numbers, with a symmetric product of the curve or of its degree-zero Picard variety according to a precise parity rule. A canonical real structure on Atiyah's indecomposable bundle gives the parallel classification of indecomposable bundles and their real loci.
A residual-third-point construction gives an exact Grothendieck-ring identity relating a cubic hypersurface to its Fano variety of lines. Under the paper's cancellation conjecture, this identity forces the Fano variety of a rational cubic to be stably decomposable and, for a cubic fourfold, birational to the Hilbert square of a K3 surface.
Beltrametti, Höring, and Novelli bound the dimension of the non-klt locus of a Fano variety by its index and classify the equality case as a generalized cone with a linear vertex. Under log canonicity they also show that a non-klt locus of the next possible dimension is a projective space or a possibly reducible quadric.
Huybrechts introduces constant cycle curves on K3 surfaces and proves that only finitely many of any fixed order occur in a fixed linear system. The proof converts torsion of the generic-point cycle into a torsion condition on a normal function and uses its non-torsion boundary class together with algebraicity of normal-function zero loci.
Near a stable bundle with vanishing Chern classes, every cohomology jump locus is the intersection of Nadel's quadratic Kuranishi cone with an explicit linear cup-product annihilator. The proof controls the full possibly nonreduced analytic germ through deformations over all Artinian local rings.
The paper reconstructs the curve and the nilpotent-cone data of a general bundle from the intrinsic Hitchin geometry of the moduli space, then classifies every automorphism as pullback, tensorization, or dualization.
Mayanskiy computes the integral Chow groups of the fixed-determinant moduli space of stable rank-two bundles on a curve through a natural split exact sequence involving projective space and symmetric powers of the curve. The argument applies blow-up and projective-bundle formulas to each wall in Thaddeus's flip sequence, telescopes the resulting relations, and then descends through a projective bundle over the moduli space.
Li and Liedtke prove that every projective K3 surface of odd Picard rank in characteristic zero contains infinitely many integral rational curves. They combine Picard-rank jumps after reduction with rigidified stable maps that can be transported through moduli.
Tziolas gives smoothing criteria for normal-crossing Fano varieties and proves that a smoothing with smooth total space exists exactly when the infinitesimal deformation line bundle on the singular locus is trivial.
Macri and Stellari realize the Fano variety of lines of a generic cubic fourfold containing a plane as a moduli space of Bridgeland-stable objects on a twisted K3 surface and relate it by one wall crossing to a moduli space of twisted sheaves.
Schwede and Smith turn Frobenius splittings into effective boundaries, proving that globally F-regular varieties in prime characteristic are log Fano and that log Fano varieties in characteristic zero have globally F-regular type.
De Fernex and Hacon prove that every simplicial complex toric Fano variety with at most terminal singularities is rigid under small projective flat deformations, by transporting its class group and polynomial Cox ring through the family.
Böhning and von Bothmer prove that the moduli space of complex plane curves of degree d is rational for all sufficiently large d, with explicit bounds in every residue class modulo three.
Tseng identifies the Chern--Schwartz--MacPherson class of a quotient-singular coarse moduli space with tangent Chern data on the inertia stack, and its stringy Chern class with the corresponding data on the double inertia stack.
Tenie proves that a Fano variety with isolated Du Bois local-complete-intersection singularities deforms to one with only 1-rational singularities; if every original singularity is 1-irrational, the variety is smoothable. Du Bois--Nakano vanishing is used to globalize simultaneous local smoothing directions.
Every projective toric variety is a fine moduli space of stable representations of a suitable bound quiver of sections. The construction recovers the chosen line bundles as tautological bundles and hinges on identifying path relations with the toric equations after irrelevant-ideal saturation.
Taking products identifies the reduced K-moduli components of two K-semistable Fano varieties with the component of their product when the factors share no simple component. The proof passes from splitting of Q-Gorenstein deformations to a finite open immersion of stacks and good moduli spaces.
The Fano index of an n-dimensional well-formed weighted projective space with canonical singularities is bounded by the sharp Sylvester-sequence quantity (s_n - 1)(2s_n - 3). The proof combines simplex barycentric inequalities, majorization, and the age criterion for canonical quotient singularities.
For a fixed-determinant moduli space of stable vector bundles on a curve of genus at least 3, every derived autoequivalence is generated by shifts, theta twists, curve automorphisms, bundle twists, and—in rank 2—dualization. The paper also constructs exact Fourier--Mukai type functors from universal extension correspondences between moduli spaces of different ranks.