Papers

We prove that if $x\in (X,\Delta,\mathbb{T})$ is a log Fano cone singularity over an uncountable algebraically closed field, and $\nu_0$ is a $\mathbb{T}$-invariant valuation with center $x$ and $A_{X,\Delta}(\nu_0)<\infty$, then the value $$ \delta(X,\Delta;\nu_0):=\inf_{\nu\in \mathrm{Val}^{\mathbb{T},*}_{X,\ni x}}\frac{A_{X,\Delta}(\nu)}{S(\nu_0;\nu)}$$ admits a minimum. The proof uses the generic limit argument. Note that there is a counterexample if we discard the log Fano cone structure.

We determine the Hilbert series of ideals generated by $d$'th powers of $n+2$ general linear forms in $n$ variables, to give upper bounds on the degree of the Hilbert series of ideals generated by $d$'th powers of $n+k$ general linear forms for $k>2$. This allows us to show that the Iarrobino-Fröberg Conjecture fails for all $n$ large enough. We also determine the degree of the Hilbert series for the ideal generated by $d$'th powers of $n+3$ general linear forms, for some values of $n$, and give counterexamples to a conjecture on the failure of the Weak Lefschetz Property for ideals generated by sufficiently large powers of general linear forms. Moreover, we determine the Hilbert series of the ideal generated by two generic quadratic forms in the exterior algebra on an even number of generators.

Let d=2g+1>4 and let A^{d-1} be the coefficient space parametrizing hyperelliptic families C_a: y^2 = x^d + a_{d-1}x^{d-1} + ... + a_1x+t over the t-line. We show that for a Zariski-generic coefficient vector a=(a_1,...,a_{d-1}) in \bar{Z}^{d-1}, for every prime p large enough, C_a has generically ordinary reduction at every prime above p. For arbitrary a in \bar{Z}^{d-1}, the set of ordinary primes of the hyperelliptic curve C: y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t over Q(a)(t) has natural density at least 1/[Q(a,\zeta_d):Q(a)]; after base change to Q(a,\zeta_d)(t), it has natural density 1.

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$. Let $A(V)$ be the moduli(Tjurina) algebra, let $A^*(V)$ be the generalized moduli algebra, and let $$ L(V)=\mathrm{Der}_{\mathbb{C}}(A(V),A(V)),\qquad L^*(V)=\mathrm{Der}_{\mathbb{C}}(A^*(V),A^*(V)) $$ be the Yau algebra and the new Yau algebra of $V$, respectively. Write $\lambda(V)=\dim_{\mathbb{C}}L(V)$ and $\lambda^*(V)=\dim_{\mathbb{C}}L^*(V)$. We prove that the difference between these two dimensions is determined by the Hessian corank. In the Morse case, the generalized moduli algebra is zero, and hence both derivation Lie algebras are zero. We have $\lambda^*(V)=\lambda(V)-1$ if $\mathrm{corank}(\mathrm{Hess}(f)(0))=1$, and $\lambda^*(V)=\lambda(V)$ otherwise. In particular, $\lambda^*(V)=\lambda(V)$ when $n\ge2$ and $\mathrm{mult}(f)\ge3$. This proves Conjecture~1.1 proposed in \cite{ChenHussainYauZuo2020}. We construct an exact sequence whose two end terms are copies of the socle of the Milnor algebra. This homological approach reveals the detailed algebraic structure underlying the dimension count. We also give an alternative proof that pushes the original arguments in the proof of \cite[Theorem~C]{ChenHussainYauZuo2020} further. The general case is obtained by combining Saito's criterion for the non-quasi-homogeneous case.

Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple-normal-crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{or}\qquad d_1=1,\ d_2\geqslant8. \] After intersecting with the standard very-general algebraic-hyperbolicity locus, we obtain Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof uses logarithmic invariant two-jet differentials, the corrected Demailly–El Goul zero-locus argument, and a two-component adaptation of slanted vector fields with base twist $\mathcal{O}_{\mathbb{P}^2}(3)$. The adapted twist three is uniform in $(d_1,d_2)$: simultaneous cancellation of the two root equations requires third Taylor differences, but no seventh-order polarization. For a linear component the third-remainder space vanishes, and ten compatible mixed generators replace the generic Taylor package. A numerical phase diagram shows that exact finite computation is needed only for the six boundary pairs $(3,3),(3,4),(2,6),(2,5),(1,9),(1,8)$. Their key vanishings are certified by full-column-rank matrices over finite fields. For two cubics, a strengthened range reduces to twelve theorem-relevant degree–twist cases; two low-weight self-tests bring the computational total to $168$ character–eigenvalue blocks. The exact source, certificates, and reproduction instructions form an accompanying computational archive. In the two-cubic case the resulting explicit estimate is \[ T_f(r)\leqslant57\bigl(N_f^{[1]}(r,C_1)+N_f^{[1]}(r,C_2)\bigr) +o(T_f(r))\ \|. \] For a conic and a quintic one may take the explicit coefficient $45$, and for a line and an octic one may take $69$.

Let $\bar{V}$ be a connected smooth complex projective variety. Let $D \subset \bar{V}$ be a normal crossing divisor. Let $\mathbb{V}$ be a complex polarizable variation of Hodge structure on $V := \bar{V} \setminus D$. Suppose that the period map of $\mathbb{V}$ is immersive at a point of $V$. We prove that for every integer $p$ with $1 \leq p \leq \dim V$, the vector bundle $\Omega_{\bar{V}}^p(\log D)$ is L-big (i.e. the tautological line bundle on $\mathbb{P}\Omega_{\bar{V}}^p(\log D)$ is big). If the local monodromy is quasi-unipotent, we give a method to determine an $m \in \mathbb{N}$ such that if $p > m$, then $\Omega_{\bar{V}}^p(\log D)$ is moreover Viehweg-big. We give the optimal value of $m$ explicitly when $V$ is a locally symmetric variety. We prove that if $V$ is a finite étale cover of the fine moduli space of smooth quintic threefolds, the result holds for $m = 90$ (in this case $\dim V = 101$). The proof of the previous results is based on a study of an augmented base locus associated with $\Omega_{\bar{V}}^p(\log D)$. In the case of locally symmetric varieties, we introduce “higher degree characteristic subvarieties”, generalizing the characteristic subvariety defined by Mok in the case $p = 1$, and prove that they coincide with these augmented base loci.

Problem F of \cite{FLOS18} asks whether an ideal generated by $r$ generic $\mu$-power forms, where $\mu\vdash d$ is a non-pure partition, has the same Hilbert series $G_{n,d,r}=\big[(1-t^d)^r(1-t)^{-n}\big]_+$ as an ideal generated by $r$ generic forms of degree $d$. We show that the answer is negative, and we determine exactly when the natural obstruction is available. Our main tool is an apolarity inclusion that converts a distinguished large part of $\mu$ into a fat-point linear system in the dual projective space. This yields a single master obstruction (Theorem \ref{thm:master}): a counterexample exists whenever, for $r$ general points of $\PP^{n-1}$, the homogeneous system $\LL(q;m^r)$ is non-empty while $\LL(q;(m-a)^r)$ has non-positive virtual dimension. We complement this with a positive result (Theorem \ref{thm:small-r}): by Stanley's theorem, Problem F has a positive answer for every $\mu$ as soon as $r\le n+1$. For ternary forms, the obstruction produces counterexamples precisely for $5\le r\le 8$, with extremal curves given by the conic through five points and products of $(-1)$-curves on del Pezzo surfaces. It is vacuous unconditionally for $r\le 4$ and $r=9$, and, assuming Nagata's conjecture, for every non-pure $\mu$ when $r\ge 10$. We also obtain counterexamples in four variables for $r=9$. An exact finite-field certificate shows that $d=14$, $\mu=(13,1)$ is the first failure for five ternary generators.

Let $X$ be a noetherian scheme with the resolution property, and let $p:Y\to X$ be a projective morphism. Suppose that $R^{i}p_{*}=0$ for $i>2$ and that $ \mathcal {O}_{X}\longrightarrow Rp_{*}\mathcal {O}_{Y} $ is an isomorphism. We show that derived pushforward induces an equivalence \[ D^{b}(Y)/\operatorname{Ker}(Rp_{*})\simeq D^{b}(X). \] As an application, we prove a characteristic-free form of the Bondal–Orlov localization conjecture for quasi-projective threefolds.

We construct a theory of Fourier transform for rational étale motives on derived vector bundles over derived Artin stacks over a field of positive characteristic, defined as the integral transform with kernel the Artin-Schreier motive. The basic properties are established: involutivity, base change, functoriality, duality, monodromicity, constructibility, and compatibilities with various natural transformations.

In this paper, we give a counterexample to the Bruzzo–Graña Otero conjecture on the curve semistability for Higgs bundles. Let $\Sigma$ be a very general smooth plane quintic curve and let $X=\Sigma^{(2)}$ be its second symmetric product. Starting from the tautological bundle $F=\mathcal{O}_\Sigma(1)^{[2]}$, we construct a rank four Higgs bundle $\mathcal{E}_s=(E_s,\theta_s)$ on $X$. Then, for every smooth projective curve $C$ and every morphism $f:C\to X$, the pullback Higgs bundle $f^*\mathcal{E}_s=(f^*E_s,f^*\theta_s)$ is semistable. However, on the other hand, $\mathrm{det}(E_s)\cong\mathcal{O}_X$ and $\int_Xc_2(E_s)=10$. Consequently, the discriminant of $E_s$ does not vanish, and $\mathcal{E}_s$ provides the desired counterexample.

We use the relative Abel–Jacobi construction of Cheltsov–Loginov–Orlov–Prokhorov to study unirational Fano threefolds. Our main result is a $50$-dimensional family of smooth unirational del Pezzo threefolds of degree one. We also construct explicit one-parameter families of terminal unirational sextic double solids and quartic threefolds of Picard rank one. Finally, we reinterpret the Roth–Iskovskikh unirationality construction for smooth double quadric threefolds in terms of a correspondence induced by the relative Abel–Jacobi map.

In this paper, we propose a method to study the Uniform Boundedness Conjecture and the Lang-Silverman Conjecture for abelian varieties $A$ defined over a global field $K$; the latter is a uniform lower bound on the heights of non-torsion rational points. Our method is inspired by Vojta's proof of the Mordell Conjecture (Faltings's Theorem). Over function fields of characteristic $0$, a recent breakthrough of Looper-Yap (arXiv:2603.23396) proves both conjectures with inexplicit bounds. In our paper, we give a new proof of both conjectures with explicit bounds, which also depend polynomially on the field $K$ unless $A/K$ admits a factor of good reduction everywhere. We also prove explicit bounds for elliptic curves over function fields of characteristic $p>0$. Over number fields, we prove both conjectures under a suitable high-dimensional Szpiro conjecture (weaker than Hindry's version, Conjecture 3.4 of https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf) that we propose.

We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over $\mathbb{F}_2$, which gives us the number of algebraic tori necessary to cover the cluster manifold over $\mathbb{F}_2$. We also show that these formulae satisfy certain recurrence relations.

In this short paper, we prove that, for any two complex elliptic curves \(E\) and \(F\), the product \(E\times F\) has degree of irrationality \(3\). The upper bound is obtained from compatible cyclic plane-cubic models of \(E\) and \(F\): three diagonal bihomogeneous sections define a dominant rational map \(E\times F\dashrightarrow\mathbb P^2\) of degree \(3\). The lower bound follows by excluding dominant rational maps of degrees \(1\) and \(2\) from an abelian surface to \(\mathbb P^2\).

Over an algebraically closed field of characteristic $p>0$, we prove that smooth Fano threefolds form 105 irreducible deformation families.

Let $0\leq k < n$, and let $\mathcal V\subseteq M_n(\mathbb C)$ be a complex linear subspace satisfying $\operatorname{rank}[S,T]\leq k$ for all $S,T\in\mathcal V$. Omladič, Radjavi, and Šivic proved the sharp bound $\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1$ and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, $\mathcal V$ consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximal-dimensional commuting subspace of $M_{n-k}(\mathbb C)$. For $n-k\geq4$, these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions $2$ and $3$, the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian. For $2\leq k\leq n-2$, we determine all of its irreducible components. If $n-k\geq4$, there are exactly two components when $n-k$ is even and exactly four when $n-k$ is odd; if $n-k\in\{2,3\}$, there are exactly two. When $n-k\geq4$, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When $n-k\in\{2,3\}$, the two components are obtained by varying the invariant $k$-dimensional subspace and the maximal-dimensional commuting subspace on the quotient, together with their transposes.

We study the $\mathbb{P}^1$-local motivic homotopy theory under the Nisnevich topology on $Sm_S$. We show that although the homotopy category $\mathcal{H}^{\mathbb{P}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable $\mathbb{P}^1$-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on $\mathbb{P}^1$. Over perfect fields, we then establish the $\mathbb{P}^1$-invariance of the higher unstable homotopy group sheaves $\pi_i^{\mathbb{P}^1}$ ($i \geq 1$) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable $\mathbb{P}^1$-connectivity theorem over fields. As a consequence, we obtain a homotopy $t$-structure on the stable $\infty$-category of $\mathbb{P}^1$-invariant Nisnevich spectra, and deduce the strict $\mathbb{P}^1$-invariance of stable $\mathbb{P}^1$-homotopy group sheaves.

Let $\mathcal{X}$ be a local orbifold Calabi-Yau threefold whose coarse space $X$ has transverse $A_{N}$ singularities along a smooth non-compact curve. We define orbifold Gopakumar-Vafa invariants, and we prove they are integers and satisfy a finiteness property. We compute our invariants for local orbifold $K3$ surfaces, where we prove an orbifold version of the classical Yau-Zaslow formula: we show that for an orbifold $K3$ surface $\mathcal S$, the genus zero Gopakumar-Vafa invariant of $\mathcal{S}$ in a class of square $2n$ is the Euler characteristic of the Hilbert scheme of $n+1$ points on the singular surface $S$.

We classify the compactified universal Jacobian spaces that are projective over the moduli stack of stable pointed curves.

We prove that if $Z$ is a tropical cycle in a tropical variety $Y$ which is algebraically trivial, then for $k\gg1$ sufficiently large, $k!\cdot Z^k\subset Y^k$ is rationally trivial. The proof involves explicitly constructing a rational equivalence $k!\cdot(p-q)^k\sim0$ in the product $C^k$, where $p$ and $q$ are any points on $C$. We use this to formulate a tropical version of Voevodsky's conjecture, provide explicit examples, and comment on potential applications to symplectic geometry.

We investigate how the complex analytic Morse Splitting Lemma relates to the generic Lê numbers of a function, and what the relation tells us about the function. As special cases, we consider functions for which the 0-dimensional generic Lê number is at most 3 and functions for which the critical locus has dimension 1.

In 2002, Lins Neto introduced three remarkable one-parameter families of holomorphic foliations by curves on the complex projective plane, of degrees two, three, and four, whose properties established that the general form of the Poincaré Problem had no solution. These foliations are known to be birationally equivalent to certain quotients of linear foliations on abelian surfaces. We give explicit formulas for these birational equivalences, obtaining, in particular, parametrizations of the leaves of the foliations. For the families of degrees three and four, we determine explicit generators for the groups of birational transformations of the projective plane preserving them. For foliations in these families admitting a rational first integral, that is, for those whose parameter is an Eisenstein rational, we give a complete description of the nature and position of the singular points of a generic integral curve, and we present an algorithm that computes the rational first integral explicitly. Lins Neto's foliations can also be defined over algebraically closed fields of positive characteristic, and, in this setting, we characterize those that are algebraically integrable. Finally, we study the integrability of the reductions to fields of positive characteristic of some of the non-integrable foliations in the complex family.

We study a restricted form of Gross's finiteness problem for algebraic minimal Calabi–Yau threefolds: those fibred by quartic K3 surfaces and realised as anticanonical hypersurfaces in stacky scrolls $\PP^1\times[\PP^3/\ZZ_n]$. Beyond the ten straight-scroll families of \cite[Table~1]{MboyaSzendroi2023}, we introduce orbifold scrolls $(\PP^1\times\PP^3)/\ZZ_n$ and apply the Reid–Shepherd-Barron–Tai criterion to determine which admit canonical anticanonical hypersurfaces. Only finitely many weight vectors arise for each $n$; we classify $n=1,2,3$ completely and compute all Hodge numbers, giving fourteen deformation families in total. Engel, Filipazzi, Greer, Mauri and Svaldi \cite{EngelFilipazziGreerMauriSvaldi2025} have since established boundedness for fibred Calabi–Yau threefolds in general, settling the existence question this restricted case exemplifies; what remains, and what we supply here, is the explicit classification: weight vectors, singularity types, and Hodge data.

Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic} \quad \Longleftrightarrow \quad \mathcal{H}^{2d} \bigl( \Sigma_\infty(X)\bigr)=0. $$ As an application, if $C_\infty(X)$ is definable in an o-minimal expansion of $(\mathbb R,+,\cdot)$ and $\dim_{\mathbb R} C_\infty(X)\le 2d$, then $X$ is algebraic.

For a flat $\mathbb{Z}_{(p)}$-scheme $X$, we introduce an obstruction class to the existence of a Frobenius lift on its reduction $X_1$ modulo $p^2$. This class is defined by replacing the cotangent complex in the classical construction of the Kodaira–Spencer class with its arithmetic analogue, the Frobenius–Witt cotangent complex. We further prove that this class coincides with the obstruction to Frobenius lifting defined by classical deformation theory. We also define its relative version and prove that this is an obstruction class to the existence of a Frobenius lift compatible with the given one on the base scheme.

We introduce the notion of universal quilt for toric CY3 singularities and construct examples when the universal quilts are themselves smooth Calabi-Yau threefolds. Such quilts are designed to contain all of the crepant resolutions of the given singularity. To obtain universal quilts we implement an algorithm that computes all triangulations for any given planar polygon. We also introduce the notion of quilt stack, to address the question of finding the most likely smooth configuration preceding a given singularity.

Let $E$ be a finite set, and let $V \subseteq \mathbb{C}^E$ be a linear subspace that is not contained in any coordinate hyperplane. The closure of $V$ in the product of projective lines $(\mathbb{P}^1)^E$ is a singular variety $Y_V$ known as the matroid Schubert variety (or arrangement Schubert variety). We use operational Chow cohomology to prove that every line bundle on $Y_V$ is the restriction of a line bundle on $(\mathbb{P}^1)^E.$ We then give combinatorial characterizations of when $Y_V$ is Gorenstein and $\mathbb{Q}$-Gorenstein, respectively. We provide examples of linear subspaces $V$ such that $Y_V$ is Gorenstein but not smooth, $\mathbb{Q}$-Gorenstein but not Gorenstein, and not $\mathbb{Q}$-Gorenstein, respectively.

We give a new intrinsic construction of the continuous tropical Prym variety, originally due to Röhrle and Zakahrov. In particular, given a harmonic double cover of graphs we explicitly construct the tropical Prym as integral torus. Moreover, we show that the tropical Prym–Torelli map from the moduli space of harmonic double covers to the moduli space of principally polarised tropical abelian varieties is continuous.

We show that border rank is equal to rank for Kruskal tensors. We also give an analogous Kruskal condition for alternating tensors, which certifies uniqueness of skew rank decompositions. Furthermore, we show that border skew rank is equal to skew rank for alternating Kruskal tensors, and we give an algorithm to find the minimum skew rank decomposition of alternating Kruskal tensors.

We give a complete characterization of the positive integer pairs $(e,m)$ that occur as the (embedding dimension, multiplicity) of a complete intersection numerical semigroup, thereby settling an open problem in the theory of relations of numerical semigroups. We prove that there exists a complete intersection numerical semigroup $\Gamma$ with $e(\Gamma)=e$ and $m(\Gamma)=m$ if and only if $(e,m)=(1,1)$ or $e\ge 2$ and $m\ge 2^{e-1}$.