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Valery Alexeev

Publications and source records attributed to Valery Alexeev.

At least 19 recordsLinked to original sources
arXiv2026

Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces

The Cassels--Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray and Voisin to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument of Ma to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels--Swinnerton-Dyer conjecture for cubic surfaces.

math.AG↗
arXiv2026

Augmentations, reduced ideal point gluings and compact type degenerations of curves

In this note we demonstrate some unexpected properties that simple gluings of the simplest derived categories may have. We consider two special cases: the first is an augmented curve, i.e., the gluing of the derived categories of a point and a curve with the gluing bimodule given by the structure sheaf of the curve; the second is an ideal point gluing of curves, i.e., the gluing of the derived categories of two curves with the gluing bimodule given by the ideal sheaf of a point in the product of the curves. We construct unexpected exceptional objects contained in these categories and discuss their orthogonal complements. We also show that the simplest example of compact type degeneration of curves, a flat family of curves with a smooth general fiber and a 1-nodal reducible central fiber, gives rise to a smooth and proper family of triangulated categories with the general fiber an augmented curve and the central fiber the orthogonal complement of the exotic exceptional object in the ideal point gluing of curves, called the reduced ideal point gluing of curves.

math.AG↗
arXiv2026

Basic properties of kappa classes

We define kappa classes on KSBA moduli stacks as classes in operational Chow cohomology. They generalize the Miller-Morita-Mumford classes on the moduli spaces of curves. We prove base-change compatibility, product and normalization formulas, and crepant functoriality, together with the vanishing of all kappa polynomials above the variation. The higher kappa classes are nonnegative on effective cycles, and the largest index of a numerically nontrivial kappa class equals the normalized variation, while the first kappa class detects the total variation. As the boundary coefficients vary, the classes are chamberwise polynomial and compatible under operational wall crossing. Finally, every positive-degree kappa class is a rational multiple of a single Chern class of a natural virtual vector bundle.

math.AG↗
arXiv2026

A tower of complete moduli spaces of Calabi-Yau $n$-folds

We construct a sequence of complete moduli spaces $$E_0 \subset E_1 \subset E_2 \subset \dots E_n \subset\dots,$$ each of which is isomorphic to a weighted projective space. These spaces parameterize certain $n$-dimensional Calabi-Yau varieties associated with the Sylvester sequence $2,3,7,43,\dots$. They generalize the moduli space of elliptic curves $\overline{M}_{1,1}=\mathbb P(4,6)$ and Brieskorn's family over $\overline{F}^{\rm BB}_{U\oplus E_8} = \mathbb P(4,10,\dotsc, 42)$, the Baily-Borel compactification of the moduli space of $U\oplus E_8$-polarized K3 surfaces. We also study fibrations in such Calabi-Yau varieties, extending to higher dimensions the theory of elliptic surfaces.

math.AG↗
arXiv2025

On lattice-polarized K3 surfaces

We propose modifications to the commonly used definitions of lattice-polarized and lattice-quasipolarized smooth K3 surfaces, collecting various versions of the definition, and determining the effects of these choices on the resulting moduli space. We fill a gap in the theory, by replacing Weyl chambers with the new notion of a ``small cone'': the true datum in the definition of lattice quasipolarized K3 surfaces. In addition, we describe the separated moduli stack and moduli space for lattice-polarized K3 surfaces with $ADE$ singularities, an important notion for applications.

math.AG↗
arXiv2025

On the moduli space of stable surfaces with $p_g=1$ realizing the minimal volume

Let $M_1$ be the moduli space of the KSBA stable surfaces $X$ of geometric genus $p_g(X)=1$ realizing the minimal possible volume $K_X^2=\frac1{143}$. We show that its reduced part $M_{1,\rm red}$ is a $10$-dimensional projective variety isomorphic to the Baily--Borel compactification $\overline{F}_Λ^{\rm BB}$ of the moduli space of $Λ$-polarized K3 surfaces, where $Λ=II_{1,9}\simeq U\oplus E_8$ is a unimodular lattice of signature $(1,9)$. By a result of Brieskorn, $\overline{F}_Λ^{\rm BB}$ is a weighted projective space. We also verify the Viehweg hyperbolicity of the base of a Whitney equisingular family of stable surfaces in $M_1$. More generally, we prove that the same results hold for the moduli space $M_c$ of KSBA stable pairs $(X,B)$ with coefficients of $B$ belonging to a set $\mathcal C\subset [0,1]$ such that $\mathcal C\cup\{1\}$ attains a minimum, say $c$, and with $p_g(X)=1$, realizing the minimal possible volume $(K_X+B)^2=v(c)$. Indeed, we show that $M_{c,\rm red}$ is independent of $c$ and that for $c\le\frac7{13}$ $M_c$ is isomorphic to $\overline{F}_Λ^{\rm BB}$.

math.AG↗
arXiv2025

The KSBA moduli space of stable log Calabi-Yau surfaces

We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi--Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking-Keel-Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov-Witten theory and mirror symmetry.

math.AG↗
arXiv2025

Non-toric brane webs, Calabi-Yau 3-folds, and 5d SCFTs

We study webs of 5-branes with 7-branes in Type IIB string theory from a geometric perspective. Mathematically, a web of 5-branes with 7-branes is a tropical curve in $\mathbb{R}^2$ with focus-focus singularities introduced. To any such a web $W$, we attach a log Calabi-Yau surface $(Y,D)$ with a line bundle $L$. We then describe supersymmetric webs, which are webs defining 5d superconformal field theories (SCFTs), in terms of the geometry of $(Y,D,L)$. We also introduce particular supersymmetric webs called ``consistent webs", and show that any 5d SCFT defined by a supersymmetric web can be obtained from a consistent web by adding free hypermultiplets. Using birational geometry of degenerations of log Calabi-Yau surfaces, we provide an algorithm to test the consistency of a web in terms of its dual polygon. Moreover, for a consistent web $W$, we provide an algebro-geometric construction of the mirror $\mathcal{X}^{\mathrm{can}}$ to $(Y,D,L)$, as a non-toric canonical 3-fold singularity, and show that M-theory on $\mathcal{X}^{\mathrm{can}}$ engineers the same 5d SCFT as $W$. We also explain how to derive explicit equations for $\mathcal{X}^{\mathrm{can}}$ using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d $\mathcal{N}=2$ theory.

hep-th↗
arXiv2025

Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism contains a curve of genus $\ge2$. For order $3$, we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain $ADE$ root lattices.

math.AG↗
arXiv2023

On the existence of ramified abelian covers

Given a normal complete variety $Y$ over an algebraically closed field $\mathbb K$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover of $Y$ branched with order $d_i$ on $D_i$. As an application, we prove that a cover of a normal complete toric variety branched on the torus-invariant divisors is itself a toric variety if the characteristic of $\mathbb K$ is equal to 0 or if the cover is Galois of degree not divisible by the characteristic.

math.AG↗
arXiv2023

Kappa classes on KSBA spaces

We define kappa classes on moduli spaces of KSBA stable varieties and pairs, generalizing the Miller-Morita-Mumford classes on moduli of curves, and compute them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient $(\mathbb P^2)^7//SL(3)$.

math.AG↗
arXiv2023

Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution

There are $75$ moduli spaces $F_S$ of K3 surfaces with a nonsymplectic involution. We give detailed descriptions of Kulikov models for one-parameter degenerations in $F_S$. In the $50$ cases where the fixed locus of the involution has a component $C_g$ of genus $g\ge2$, we identify normalizations of the KSBA compactifications of $F_S$ via stable pairs $(X,εC_g)$, with explicit semitoroidal compactifications of $F_S$.

math.AG↗
arXiv2023

Compact moduli of K3 surfaces

We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces, in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface $(X,L)\in F_{2d}$. The main new notion is that of a recognizable divisor $R$, a choice which can be consistently extended to all central fibers of Kulikov models. We prove that any choice of recognizable divisor leads to a semitoroidal compactification of the period space, at least up to normalization. Finally, we prove that the rational curve divisor is recognizable for all degrees.

math.AG↗
arXiv2023

Stable pair compactification of moduli of K3 surfaces of degree 2

We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,εR)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.

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