Papers

We completely characterize when a general curve of genus $g$ admits a non-degenerate degree $d$ map to projective space $\mathbb{P}^r$, that is $k$-secant along a linear space $\mathbb{P}^s \subset \mathbb{P}^r$ and deforms in a smooth family of expected dimension. This gives an optimal improvement of a theorem of Farkas, who gave necessary conditions, and proved sufficiency under additional numerical assumptions. Our main theorem shows that the natural necessary conditions are in fact sufficient, yielding a generalization of the classical Brill-Noether existence theorem. The proof relies on deformation theory of stable maps, and is valid in arbitrary characteristic.

For every integer $n\ge3$, we construct a smooth projective manifold $X$ of complex dimension $n$ whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every Kähler metric has negative total scalar curvature. In particular, $X$ admits no Kähler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a Kähler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.

We characterize Griffiths positivity of vector bundles on smooth projective varieties by a finite-dimensional convex condition on first jets of sections of positive twists. A nonzero positive matrix-valued measure annihilated by the adjoint Levi operator gives the dual obstruction. We also give an explicit bundle on an abelian surface that satisfies the first-level condition, but whose complete untwisted evaluation has a differential kernel.

Tropicalizations of Grassmannians resp. flag varieties parametrizing (flags of) linear spaces have been studied intensely and reveal rich combinatorial structure. Linear degenerations of flag varieties provide ways of interpolating between products of Grassmannians and flag varieties. Tropical versions of these have been introduced recently, but their combinatorial description and computation remains a challenge. We consider a particular case which allows the use of matroidal methods, namely linear degenerate short flag varieties parametrizing tuples of linear spaces such that a projection of one is contained in the other. Their fibers turn out to be linear spaces themselves under mild assumptions, which allows a comprehensive combinatorial description of their tropicalizations, and accordingly, of tropicalizations of linear degenerate short flag varieties.

We construct an irrational one-point Seshadri constant on the blow-up $X_9$ of $\mathbb{P}^2$ at nine very general points. The divisor $L=9H-3(E_1+\cdots+E_5)-2(E_6+\cdots+E_9)$ is ample and satisfies $\varepsilon(L;x)=2\sqrt{5}$ at a very general point $x\in X_9$. To prove this, we establish that the Seshadri constant of $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(1,1)$ at ten very general points is $1/\sqrt{5}$, completing the reflection approach proposed by Dionne and Roth for the ten-point case. We also prove that the same equality holds at a very general free orbit of a fixed dihedral group of order ten. This yields an irrational one-point Seshadri constant on the singular quotient surface. A plane model of its minimal resolution and a deformation of the blow-up centers then give the result on $X_9$.

We construct a rank-two ample holomorphic vector bundle on an abelian surface that admits no smooth Griffiths-semipositive Hermitian metric, thereby giving a counterexample to the Griffiths conjecture. The proof combines a curvature obstruction that persists under deformations with the openness of ampleness in an irreducible moduli space of stable sheaves.

We provide the first compact examples of type-changing generalized Calabi–Yau manifolds using Fei's construction of non-Kähler Calabi–Yau manifolds on branched covers of twistor spaces \cite{Fei16}. The structure, which is generically of type 1 and changes to type 3, is constructed on the product of a hyperelliptic surface of genus three and a four-torus, $\Sigma_3 \times \mathbb{T}^4$, with starting point a non-Kähler Calabi–Yau structure.

Let $f=(f_1,\dots,f_r)$ be a complete factorization of a central hyperplane arrangement $D$ in $X=\mathbb{C}^n$. For a monoid ideal $K\subseteq \mathbb{N}^r$ we study the Bernstein-Sato ideal $B^K_f$ of $f$ along $K$, that is, the $\mathbb{C}[s]$-annihilator of $\mathcal{D}_X[s]f^s/\sum_{m\in K}\mathcal{D}_X[s]f^{s+m}$. When $D$ is free we compute two families of these ideals with the help of AI. For the unit shift $K=\langle e_i\rangle$ we prove that $B^{-e_i}_f$ is generated by an explicit product of linear forms indexed by the dense edges of $D$ contained in $D_i$. This determines all the Bernstein-Sato ideals $B^{a,b}_f=\operatorname{Ann}_{\mathbb{C}[s]}\mathcal{D}_X[s]f^{s-a}/\mathcal{D}_X[s]f^{s-b}$, $a\geq b$, of a free arrangement, generalizing formulas of Maisonobe (2016) and Bath (2020). The main new ingredient identifies the multiplicities of the relative characteristic cycle of $\mathcal{D}_X[s]f^s/\mathcal{D}_X[s]f^{s+e_i}$ along the conormal bundle of the origin with the coefficients of the Hilbert series of an Artinian complete intersection attached to a generic Ziegler restriction of $D$; the total multiplicity computed in Wu (2022) then forces all the resulting coefficientwise upper bounds to be equalities. For the coordinate monoid ideal $K=\langle e_1,\dots,e_r\rangle$ we show that $B^K_f$ is generated by one Euler relation for each irreducible factor of the essential quotient of $D$. Finally, we show that the zero locus of a Bernstein-Sato ideal along a monoid ideal need not be a finite union of translated linear subvarieties, even for a reduced free arrangement in $\mathbb{C}^2$: for $f=(x,y,x+y,x+2y)$ and $K=\langle 3e_1,3e_2\rangle$ we compute $B^K_f$ exactly and find an irreducible quadric component. This disproves a conjecture due to Budur.

We prove a geometric detection theorem for crystalline local systems on the generic fiber of a smooth proper scheme over the ring of integers of a $p$-adic field. The special-fiber crystalline realization detects geometric absolute irreducibility and, modulo constant rank-one twists, determines the restriction to the geometric fundamental group. The main ingredient is a determinant argument combining a finite-order rank-one reduction, period comparison, and the Plücker embedding. As an application, using Abe–Esnault's finiteness theorem and finiteness of stable lattices, we prove finiteness, up to arithmetic character twists, of geometrically absolutely irreducible crystalline local systems of fixed rank. Coefficients may lie in any fixed finite extension of $\mathbb Q_p$, and no bound on the Hodge–Tate weights is imposed.

If $X$ is smooth projective complex variety, $D$ is a simple normal crossings divisor and $A$ an ample divisor with support $D$, then there is an associated filtration of the Lie algebra of on the affine variety $U=X\setminus D$ induced from the subspaces $T_X(\log D)(nA)\subset T_U.$ We show that the associated graded Lie algebra is finitely generated in a strong sense. We deduce as a consequence that vector fields on any smooth affine variety form a finitely generated Lie algebra.

We propose a bounded degree SOS+SONC hierarchy for constrained polynomial optimization, termed B-SOS+SONC. Starting from Lasserre's bounded-degree SOS framework, we enlarge the certificate cone from SOS to the recently introduced SOS+SONC cone, thereby combining the algebraic strength of semidefinite relaxations with the sparse structure captured by circuit polynomials. We show that, for each fixed certificate degree, the resulting hierarchy is complete, that is, its optimal values are monotone and converge to the global optimum. Moreover, we derive an explicit SDP-REP reformulation, so that each relaxation can be solved within a tractable convex optimization framework over semidefinite and relative entropy cones. Beyond the optimization hierarchy itself, we investigate structural properties of the SONC cone and introduce the notions of first-order and second-order SONC-convexity. This leads to a new sufficient condition for first-level exactness of the B-SOS+SONC hierarchy. Numerical experiments illustrate that the proposed hierarchy often yields tighter lower bounds than the B-SOS relaxation while remaining tractable.

We construct the moduli space of marked polystable $\mathrm{SL}_r(\mathbb C)$-Higgs bundles via analytic methods. The resulting space is normal and Hausdorff, and its local models are products of a Teichmüller neighbourhood with a quotient of a quadratic cone. These models preserve the curve parameter and identify the stabilizers. We prove agreement with the complex and unitary gauge-quotient topologies and establish the coarse moduli property for semistable families over reduced analytic bases. The mapping class group and Higgs scaling act holomorphically. Besides, on the stable locus, the moduli spaces that we consider agree with the joint moduli constructed by Collier, Toulisse and Wentworth.

Geometric Goppa codes are constructed by evaluating elements of an algebraic function field at the places of degree one. Extending this, generalized geometric Goppa codes were defined in [XNL99] and [Hey02] by allowing evaluation at places of arbitrary degree. In this paper, we establish necessary and sufficient conditions for equality of generalized geometric Goppa codes, following the approach of [MP93]. These conditions serve as a key tool in our characterization of iso-dual and self-dual generalized geometric Goppa codes. Given a separable extension M/F of algebraic function fields and an iso-dual generalized geometric Goppa code over F, we extend the construction of [CPQT24] to obtain iso- dual generalized geometric Goppa codes over M. Furthermore, we study self-duality of generalized AG codes ([XNL99], [DM03]).

We prove an $S$-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let $C$ be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-$p$ positive integers $m_i$ and any finite set of closed points $S\subset C$, we prove that $C\setminus S$ meets the special divisor $Z(m_i)$ for all but finitely many $i$. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford–Tate conjecture.

We recast the Lange-Stuhler theorem as the universal property of a $2$-cartesian square of classifying stacks. This formulation shows that every Frobenius-periodic vector bundle of rank $r$ on an arbitrary fibered category in characteristic $p>0$ is trivialized by a finite étale torsor, without geometric hypotheses or restrictions on the ground field beyond its characteristic. The construction extends from $\mathrm{GL}_n$ to group schemes over finite fields that are connected affine or of finite type.

We prove that every Deligne–Mumford superstack over $\mathbb{Z}[1/2]$ with finite inertia admits a coarse moduli superspace. This generalizes the Keel–Mori theorem to the super setting. As part of our argument, we introduce a notion of integral morphisms in algebraic supergeometry and use it to establish an effective descent result for étale morphisms of algebraic superspaces. We also include an example of a non-Deligne–Mumford superstack with finite inertia that does not admit a coarse moduli superspace.

We prove the existence of weight 2 modular forms for some Veech groups, whose Fourier coefficients at a cusp determine the counts of numbers of branched Galois covers of punctured elliptic curves. We further prove there exist cusp forms, counting ramified genus $2$ covers, that are not invariant under any congruence subgroup of SL(2,Z). The main results may be viewed as analogs of the construction of an elliptic curve obtained as the quotient of the Jacobian of a classical modular curve, whose point counts modulo primes determine the coefficients of a Hecke eigenform, but where the uniformizing groups may be noncongruence.

We prove that the principal minors of an orthogonal projection matrix determine the coordinatewise absolute value of its image. More precisely, if $W_1,W_2\subset \mathbb{C}^n$ are subspaces whose orthogonal projection matrices have equal corresponding principal minors of all orders, then $|W_1|=|W_2|$, where $|W|$ is the image of $W$ under the coordinatewise absolute value map. The proof is divided into different cases related to the connectivity of the associated matroids. We provide a stratification of the Grassmannian in terms of the connectivity levels and give a structure theorem for the elements of each stratum.

We prove the Eisenbud–Huneke–Ulrich conjecture on powers of m-primary ideals with partially linear minimal free resolutions in characteristic zero.

We describe a polytopal realization of categorical $n$-associahedra. The normal fan of this polytopal realization is a modification of the authors' velocity fan and was found by OpenAI's Astra model.

In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold $Z = {\rm{Tot}}({\bf{K}}_{X})$, where $X$ is a rational homogeneous variety. Since $X$ is a Fano variety, $Z$ is a resolution of the singularity at the vertex of the affine cone ${\rm{Aff}}(X)$, provided the cone is built using the anticanonical polarization ${\bf{L}} = {\bf{K}}_{X}^{-1}$. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on $Z$, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over $Z$ admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.

This article is the first of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In this article, we generalize a local version, due to Shiho, of the Cartier transform to log smooth schemes. More precisely, let $k$ be a perfect field of positive characteristic and equip $\operatorname{Spec}k$ with the trivial logarithmic structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ where $S$ is log flat and locally of finite type over $\operatorname{Spec}k,$ we obtain, under the assumption that the exact relative Frobenius lifts over the Witt vectors of $k$ to a morphism between log smooth schemes over $S,$ a fully faithful functor from the category of quasi-coherent modules on the base change $X'=X\times_{S,F_S}S$ of $X$ by the Frobenius $F_S$ of $S,$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. For this, we prove a log flat descent theorem for morphisms, based on previous work by Kato.

We prove the remaining positive characteristic cases of the Borisov-Alexeev-Borisov conjecture for geometrically integral $\varepsilon$-klt del Pezzo surfaces over arbitrary fields. Bernasconi and Martin established boundedness outside characteristics $2$, $3$, and $5$, and reduced the remaining cases to a uniform bound for the irregularity. We prove this bound, thereby completing the boundedness theorem in all positive characteristics.

In 1959, Snapper showed that the Euler characteristic of the tensor powers of a line bundle on a normal projective scheme is a polynomial, later named the Snapper polynomial. Positivity of coefficients of Snapper polynomials implies various notions of positivity of line bundles, which we study through the lens of magic positivity and real-rootedness. We introduce zonotopal classes in the Grothendieck $K$-ring of vector bundles of the toric variety for any loopless matroid, and prove that their Snapper polynomials are magic positive. Our proof realizes such a Snapper polynomial as a weighted independence polynomial of the Dilworth truncation along certain lines of the matroid. As a consequence, their coefficients are positive, and their $h^{\ast}$-polynomials are real-rooted. In the realizable case, this polynomial is the multigraded Hilbert polynomial of the wonderful variety embedded in a product of projective lines. We introduce analogous line bundles on the Deligne–Mumford–Knudsen moduli space $\overline{\mathcal M}_{0,n}$ and prove that their Snapper polynomials are magic positive. For cotangent line bundles whose first Chern classes are distinct $\psi$-classes, which are not zonotopal, we nonetheless prove that their $h^{\ast}$-polynomials are real-rooted, whereas their Snapper polynomials are magic positive if and only if $n\leqslant7$. More generally, we introduce saturated and weakly saturated $K$-classes of matroids, which furnish a sufficient and a necessary condition for magic positivity of Snapper polynomials in terms of their dragon Hall–Rado polymatroids.

Let $C\subset \PP^2$ be a reduced irreducible complex plane curve with complement $U$. In this paper we prove that $\chi(U)\geq \nu(C)$, where $\nu(C)$ is the freeness defect of $C$ defined in \cite{Dim24}. When $C$ is a rational curve with unibranched singularities, this inequality proves the conjecture of Dimca and Sticlaru, that a rational unibranched plane curve is either free or nearly free.

Consider a number field $K$, a polarized endomorphism $f:X\to X$ on a normal projective variety, and a non-exceptional point $\alpha$ for $f$. We show that the extension $K(f^{-\infty}(\alpha))/K$ generated by the backward orbit of $\alpha$ is virtually abelian if and only if $f$ is an exceptional map with CM and $\alpha$ is $f$-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over $\mathbb{C}$ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.

The cotangent space to any subvariety of a Grassmannian is naturally identified with a space of linear homomorphisms. We use this to define a new statistic on a subvariety of a Grassmannian, called conormal rank. We show that Chow–Lam forms and their natural generalizations have low conormal rank, and moreover that hypersurfaces of sufficiently low conormal rank are all of this form, generalizing a result of Gelfand–Kapranov–Zelevinsky. We also compute the conormal rank of many families of varieties, including Schubert varieties, torus orbit closures, and positroid varieties.

We compute an explicit contribution of supercuspidal representations arising from Yu's construction to the middle étale cohomology of local Shimura varieties. For the proof, we construct special affinoids in local Shimura varieties whose reductions are isomorphic to Heisenberg Deligne-Lusztig varieties and then apply positive-depth Deligne-Lusztig theory. Along the way, we propose a stacky analogue of special affinoids using the stack of local shtukas and observe that the `general canonical subgroup problem' can be solved purely on the stack of local shtukas, which yields the corresponding claim on Shimura varieties via the Hodge-Tate period map.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(g-1,g-1)^{\mathrm{hyp}}$ for $g>2$. These curves are precisely those generated by the Veech $2p$-gons, where $p=2g+1$ is prime.

Given a smooth projective curve $X$ over $\mathbb{C}$, we show that the twisted character variety of $X$ is equivalent to a component of the character variety of a stacky curve $\mathcal{X}$, constructed by adding an orbifold point to $X$. This allows to interpret the non abelian Hodge correspondence in degree $d$ over $X$ as a statement about the geometry of the Hodge moduli space of $\lambda$-connections on $\mathcal{X}$. Following Shende's approach (see arXiv:1411.4975), we exploit this interpretation to compute the weights of the tautological classes on the cohomology of the twisted character variety, by replacing $\mathcal{X}$ with a simplicial scheme $\Delta_{\mathcal{X}}$ obtained from a triangulation.