Daily Math Paper Digest
Miscellaneous topics in algebraic geometry.
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.