Algebraic Geometry Papers

ALT In this paper, we develop an explicit calculation of ordinary topological Euler characteristics of the coarse moduli spaces underlying Hassett spaces in the minimal all-light chamber and the Deligne-Mumford compactification. For genus g≥2, we organize the calculation through a decomposition indexed by stabilization graphs and their decorations. The Burnside-Lefschetz formula allows us to compute the resulting contributions, taking into account graph symmetries and their fixed loci. The calculation in genus two illustrates the method explicitly and, in the Deligne-Mumford case, recovers the generating function of G. Bini, G. Gaiffi, and M. Polito (arXiv:math/9806048), with the corrections of G. Bini and J. Harer (arXiv:math/0506083), providing a check on the construction. We also carry out the calculation in genus three in an appendix. Together with an elementary genus-one calculation based on collision partitions and rational necklaces, these formulas yield complete asymptotic expansio

ALT We study G-Mori fibre space models of G-del Pezzo surfaces, focusing on surfaces over non-closed fields. We construct explicit examples of birationally rigid sextic del Pezzo surfaces and give a negative answer to a generalization of Kollár's question on birational rigidity and field extensions: a sextic del Pezzo surface of Picard rank 1 can fail to be birationally rigid over its ground field but become birationally rigid after an algebraic field extension. We interpret this phenomenon in terms of the failure of injectivity of natural maps in nonabelian Galois cohomology induced by the inclusion of automorphism groups into birational automorphism groups. We also show that finite pliability is unbounded: for every integer d≥slant2, we construct a sextic del Pezzo surface of index 3 over a field of characteristic zero whose pliability is exactly 3ᵈ+1.

ALT Let G be a connected reductive group with Borel subgroup B, flag variety ℬ=G/B, and Weyl group W. We define the Hessenberg functor on the geometric Hecke category by microlocalizing along the diagonal in ℬ×ℬ and pushing forward along the Springer map. It induces the Hessenberg character of the Iwahori–Hecke algebra, with values in graded W-representations. On Kazhdan–Lusztig basis elements indexed by smooth elements of W, this character recovers the cohomology of the associated regular semisimple Hessenberg varieties. We identify the Hessenberg functor with the microlocalization of Lusztig's character functor and compute the Hessenberg character using Lusztig's nonabelian Fourier pairing. More generally, for families of Hessenberg schemes defined by pure-dimensional B-stable closed cones in the tangent space T_eBℬ, we show that monodromy on the intersection cohomology of regular semisimple fibers factors through W. When the cone is the tangent cone of a Schubert variety, we give a loca

ALT A prolongation variety of a polynomial differential system S is an algebraic variety defined by the equations of S and their derivatives up to a given order. We compute the geometric degree of the prolongation varieties associated with the system x₁'=x₂², x₂'=x₁² in terms of the prolongation order, thereby proving a conjecture stated by Gleb Pogudin in Section 2, Question 6 of "Some Open Problems in Differential Algebra". Our approach is motivated from the beginning by doing computational experiments in OSCAR in order to find the number of solutions of a zero-dimensional system using algebraic Thomas decomposition. The result will follow from a coprimality condition on a family of polynomials with integer coefficients, which is verified by showing that their roots satisfy a strong interlacing property. Finally, we propose an extension of the conjecture for the systems x₁'=x₂ⁿ, x₂'=x₁ⁿ with n>2.

ALT We determine the universal constant in the large-genus asymptotic expansion of super Weil-Petersson volumes. The argument adopts the uniform-transport strategy for shifted theta intersection numbers. The uniform large-genus theorem for normalized BGW numbers of Guo-Norbury-Yang-Zagier determines the amplitude at a simpler endpoint. The result is C=π⁻⁹/². We give a proof of the Griguolo-Papalini-Russo-Seminara conjecture for fixed boundary lengths and fixed descendant degrees. The boundary formula is uniform when the sum of the lengths is o(√ g), and we exhibit an obstruction at the √ g scale. We also compute the first four inverse-genus coefficients by two exact recursive methods and derive general polynomial structure results. Finally, we show that, for each fixed genus, the super Weil-Petersson volume divided by n! is a polynomial in the number n of marked points.

ALT In this article, we use Wu's method to prove Budur's multi-variable n/d-conjecture for completeness.

ALT We provide an algorithm for enumerating ordinary K3 surfaces over a fixed finite field based on Taelman's theorem. We use this algorithm to prove that there exist no K3 surfaces without rational points over Fq for every prime power q≥ 13. For elliptic K3 surfaces, we give a geometric proof of this non-existence without computer assistance. On the other hand, we also provide explicit pointless K3 surfaces over Fq for every prime power q≤ 11.

ALT We consider the comparison between a Milnor fibre of a function f and its section by the zero locus of a second function s in a general ambient variety X. We give a formula that generalizes those of Lê and Siersma, expressing the difference of their Euler characteristics as a sum of local contributions associated with the critical points of the restriction of s. We provide an algebraic formula in the case that X is smooth of pure dimension outside f⁻¹(0), and show some applications and examples.

ALT We prove that a finite surjective post-critically finite self-map of a variety in characteristic zero remains post-critically finite on every preperiodic integral effective divisor after passing to its periodic image with its return map. Iterating this result proves PCF all the way down. A detailed report on AI usage is contained in the manuscript.

ALT A ℚ-homology quadric is a smooth compact complex surface which has the same ℚ-homology as a quadric surface, that is, the product of two 2-dimensional spheres. It is called fake if it is minimal and not rational. Among them are the fake quadrics which are isogenous to a product of complex curves. In this paper, we determine the parity of the intersection forms of all fake quadrics isogenous to a product of unmixed type. In particular, we show that there are fake quadrics with odd intersection forms.

ALT For every ideal of minors of a generic complex matrix, we determine all roots and their multiplicities in the Bernstein-Sato polynomial. These roots lie in explicit rational intervals indexed by the rank strata. Every nonintegral root is simple, whereas the multiplicity of each integral root equals the number of labels producing that root. We deduce the strong motivic monodromy conjecture, including the inequality relating pole orders to root multiplicities; for minors of size r, the largest possible root multiplicity over all matrix dimensions is ⌊√(4r-1)⌋. Keywords. Bernstein-Sato polynomial, determinantal ideal, root multiplicities, strong monodromy conjecture.

ALT Over ℂ, we compute the integral Chow ring of A₃, the moduli stack of principally polarized abelian threefolds. Along the way, we compute the integral Chow ring of ℳ₃, the moduli stack of smooth genus-three curves. The ring CH(A₃) is generated by the Chern classes of the Hodge bundle, along with three classes supported on the locus of decomposable abelian varieties, subject to 16 relations. The ring CH(ℳ₃) is generated by the Chern classes of the Hodge bundle, with five relations. In particular, we find that the top Chern class of the Hodge bundle, λ₃, has order 504 on A₃ and 252 on ℳ₃. For A₃, this removes the factor of 4 from the bound 252midord(λ₃)∣ 1008 of Ekedahl and van der Geer.

ALT Let X⊂PP^M-1 be a smooth hypersurface of degree d<M, and let \Ku(X) be its Kuznetsov component, the orthogonal complement of the line bundles OO_X,…,OO_X(M-d-1) in the derived category. We prove that \Ku(X) determines X for every smooth quartic in PP⁴k+1, k≥1; together with earlier work, this gives categorical Torelli for every smooth hypersurface of degree d<M with d∤ M. The same algebraic reconstruction proves the classical generic Torelli theorem for these quartics, which completes generic Torelli for smooth hypersurfaces whose degree does not divide M, cubic surfaces excepted. When d divides M, the Kuznetsov component is a Calabi–Yau category, as for cubic fourfolds, where categorical Torelli is known to fail. We show that, apart from cubic fourfolds, such a hypersurface has a number of Fourier–Mukai partners bounded by a constant depending only on d and M, and that categorical Torelli holds in sufficiently large degree, in every dimension. The main tool is a deformation theorem: w

ALT Let X be a smooth complex projective variety and D a reduced simple normal crossings divisor. We prove that, given a complete curve of negative (K_X+D)-degree, through its general points there is a rational curve whose normalization meets D in at most one point. The proof uses ordinary weighted jets of a smooth family constructed from (X,D), a positive parameter line bundle, and an intersection calculation on a ruled surface. We obtain an explicit degree bound depending on a projective embedding. We apply this construction to a cyclic cover compactification associated with a smooth Campana orbifold (X,∑ᵢ(1-1/mᵢ)Dᵢ). If its anticanonical divisor is big, we show that there is a covering family of free divisible Campana rational curves with fixed numerical class and contact data. In Picard number one case, the orbifold is Campana rationally connected.

ALT We construct a cohomology theory H_A(-) canonically attached to any 1-dimensional commutative group scheme A over a connective E∞-ring. The additive group in characteristic zero gives 2-periodic de Rham cohomology, the multiplicative group gives topological periodic cyclic homology, and an elliptic curve gives elliptic cohomology. The construction extends to the G-equivariant setting for an algebraic group G, yielding in the elliptic case a theory of equivariant elliptic cohomology on algebraic varieties with G-action that takes values in sheaves on the stack of G-bundles. Over the complex numbers, and for G connected reductive, this lifts Grojnowski's theory from the moduli space of G-bundles to the stack. The key player is the category of perfect complexes on A with 0-dimensional support, equipped with the convolution product induced by the group law of A. It plays the role of a categorified circle and is used to define a loop space L_Aᴳ(X), whose structure sheaf gives rise to H_Aᴳ(X

ALT We study a bounded-degree form of a motivic Vojta inequality for pure André motives, arising from a question of Kato. We verify this inequality for infinitely many non-CM elliptic curves with unbounded stable Faltings height. The point is that ramification at the finite poles of the j-invariant gives the required discriminant bound.

ALT We prove Fujita's advanced generalized Mukai conjecture for smooth complex Fano varieties. Our starting point is the existence of independent two-point Gromov–Witten correspondences. We show that there are finitely many unsplit families of rational curves whose chains connect X. We use these families to bound the anticanonical degrees throughout the successive minimal horizontal construction, so every family in that construction is unsplit. Fujita's contraction theorem and inductive classification then give the required list of varieties.

ALT We construct numerical Bridgeland stability conditions on the category of graded matrix factorizations of every smooth complex projective hypersurface of degree d≥2. These conditions are of Gepner type (τ,2/d), where τ is the grading shift autoequivalence. By Orlov's theorem, this gives stability conditions of Gepner type on all Calabi–Yau hypersurfaces and on Kuznetsov components of all Fano hypersurfaces. These conditions satisfy a mass-Hom bound and the support property on the full numerical Grothendieck group. Semistable objects of fixed numerical class and phase admit proper good moduli spaces.

ALT Using adelic intersection theory and following the method envisioned by Parshin and Moret-Bailly, we prove a conjecture of Vojta for curves of genus two and a weak form of the abc conjecture

ALT In this note we prove a simple generalization of cohomology and base-change which seems to be missing from standard sources. We illustrate its usefulness by providing an alternative proof of a result of Ishimura on descent of vector bundles along blowups at a smooth center.