Papers

Let $\mathcal{P}_{n,d}$ be the space of polynomials in $n$ variables over $\mathbb{F}_q$ of degree at most $d$. Two polynomials $f,g\in\mathcal{P}_{n,d}$ intersect if $f(\mathbf a)=g(\mathbf a)$ for some $\mathbf a\in\mathbb{F}_q^n$. A star consists of all polynomials $f\in\mathcal{P}_{n,d}$ satisfying $f(\mathbf a)=b$ for fixed $\mathbf a\in\mathbb{F}_q^n$ and $b\in\mathbb{F}_q$. We completely classify the maximum intersecting families in $\mathcal{P}_{n,d}$. When $n=1$ and $d\geq 2$, it was previously shown that all maximum intersecting families are stars. We prove that the same conclusion holds for all $n\geq 2$ and $d\geq 2$ when $q$ is odd. When $q$ is even, however, the situation is more subtle, and a new phenomenon emerges: for $q\geq 4$, maximum non-star examples exist precisely when $d\leq n$. Along the way, we prove two further results of independent interest. First, we determine the span of nonvanishing polynomials in $\mathcal{P}_{n,d}$. Second, we characterize all linear functionals $\Psi\colon\mathcal{P}_{n,d}\to\mathbb{F}_q$ whose kernels are disjoint from the set of nonvanishing polynomials. The first result plays a crucial role in the proof of our main result; the second is a Gleason–Kahane–Żelazko theorem for polynomials of bounded degree over finite fields.

Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.

In this article, we show that several candidate motivic (unipotent) nearby functors coincide on perverse Nori motives. In particular, the canonical functor defined via the universal abelian factorization admits an expression in terms of the six operations and yields a monodromy sequence on perverse Nori motives. We then deduce the motivic integral identity of Kontsevich-Soibelman for perverse Nori motives. In the appendix, we also prove a version of Beilinson's equivalence for perverse sheaves of geometric origin, which is used repeatedly throughout the article.

In this note, the second author's quantized volume comparison conjecture is solved: If $X$ is a $K$-semistable Fano manifold of dimension $n$, then for every integer $m\geq1$, \[ h^0(X,-mK_X)\leq h^0(\mathbb P^n,-mK_{\mathbb P^n}) =\binom{n+m(n+1)}{n}, \] and equality for one $m$ characterizes projective space. Somewhat surprisingly, the same statement actually holds whenever $T_X$ is slope semistable with respect to $-K_X$. The idea is to apply a jet-dimension counting trick to a filtration of subsheaves induced by $H^0(X,-mK_X)$. Then the slope semistability condition yields the desired dimension bound.

We prove an asymptotic structural property of generic initial ideals. This single phenomenon yields results on Hilbert functions, persistence, hyperplane restriction, graded Betti numbers, combinatorial shadow minimization, and Lefschetz properties.

We prove that any two smooth toric Fano fourfolds whose integral cohomology rings are isomorphic as graded rings are diffeomorphic.

Let $\mathcal O$ be a nonzero nilpotent orbit in a complex simple Lie algebra $\mathfrak g$. For $\mathfrak g$ of classical type, we classify the orbit closures $\overline{\mathcal O}$ that are isomorphic to $(T^*X)^{\mathrm{aff}}$ for a smooth quasi-affine variety $X$. In types $G_2$, $F_4$, and $E_8$, we show that no nonzero nilpotent orbit closure admits such a cotangent model.

Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\K$ be an infinite field and $a_1,\ldots, a_k;$ $b_1,\ldots , b_k$ be two sequences of different points in~$\K^n$, $n\ge 2$. For any sequence $L_1,\ldots, L_k\in SL(n,\K)$ there exists a polynomial automorphism $\Phi: \K^n\to \K^n$ with jacobian one such that $\Phi(a_i)=b_i$ and $d_{a_i} \Phi= L_i$ for $i=1,\ldots, k.$

We find all possible K-polystable degenerations of smooth K-polystable Fano threefolds in the family \textnumero2.21, and describe the component of the moduli space $M^\text{K-ps}_{3,28}$ which parametrises this family.

A multiplicity-free subvariety of a product of projective lines is a subvariety with the property that the expansion of its Chow class in the usual basis has all coefficients equal to 0 or 1. We show that, over a field of positive characteristic, a multiplicity-free subvariety has F-rational singularities. This strengthens Brion's result that multiplicity-free subvarieties are normal and Cohen–Macaulay, and that they have rational singularities over a field of characteristic 0.

In this work, we compute the operator product expansion density for the genus-two handlebody by exploiting its duality with extremal conformal field theories arXiv:0710.2129. Using a Poincaré construction, we analytically determine the first- and second-order corrections in the pinching parameter to the genus-two OPE density through a direct inversion of the partition function. As a consistency check, we also compute the genus-one density within the same Poincaré framework and find perfect agreement with the result obtained from the lightcone bootstrap analysis arXiv: 1906.04184. It is known that, at genus one, the OPE density of extremal CFTs exhibits a negativity pathology for states lying just above the black-hole threshold. We show that this pathology continues to persist at genus two, despite the emergence of a nontrivial inversion kernel. In particular, the OPE density associated with states of both odd and even spin $j_i$ remains negative within a very narrow band above the black-hole threshold. We further analyze the sharp transition from large positive to large negative values of the OPE density across a curve along which the density vanishes.

We study the relation between linear stability of generated linear series on smooth curves and slope stability of their associated syzygy bundles. Motivated by conjectures of Mistretta and Stoppino, we establish new cases in which linear stability implies slope stability, focusing first on generated linear series over general curves and then on curves lying on polarized K3 surfaces. In the case of general curves, we use Brill-Noether-theoretic arguments to relate the numerical conditions on the linear series to the semi-stability of the syzygy bundle. For curves on K3 surfaces, we combine Lazarsfeld-Mukai bundles with Bridgeland stability conditions and restriction techniques to obtain slope-stability results under explicit degree bounds. These results provide further evidence for the expected equivalence between linear stability of linear series and slope stability of syzygy bundles.

We propose a refinement of the quantum Hikita conjecture of Kamnitzer, McBreen, and Proudfoot that bridges the representation theory of Coulomb branches with the enumerative geometry of Higgs branches. We also introduce a general framework for proving it, which we carry out for ADE quiver gauge theories with minuscule framings and for the gauge theory corresponding to the Jordan quiver. As an application, we use the resulting quantum Hikita isomorphisms to give a geometric description of graded traces on quantized Coulomb branches.

This is the first paper in a series on the Gromov-Witten theory of the universal abelian variety over the moduli space of principally polarized abelian varieties of dimension $h$. We conjecture that the generating series of Gromov-Witten classes, when summed over the degree against the principal polarization, is a cycle-valued quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and satisfy a holomorphic anomaly equation. These conjectures generalize the quasimodularity of the Gromov-Witten theory of elliptic curves to higher dimension and raise interesting questions regarding enumerative mirror symmetry for abelian varieties. In genus $1$ it specializes to a conjecture of Greer and Lian which was proven by Iribar Lopez after tautological projection. We also discuss a special family of abelian varieties with a conjectural relation to Siegel quasimodular forms of higher genus. The main result of the paper is a proof of the conjectures in genus $2$ after tautological projection. For that we introduce quotient Gromov-Witten invariants which are indexed by the characteristic polynomial of the curve class and are shown to determine all descendent Gromov-Witten invariants satisfying a degree conditions. We then give an explicit formula for all genus $2$ quotient invariants after tautological projection as the Shimura lift of the product of two Eisenstein series. The formula is based on a curious modular identity derived in a joint appendix with Brandon Williams.

We answer a question of Li–Liu–Xu: every $n$-dimensional klt germ $x\in(X,\Delta)$, where $n\ge2$, satisfies the sharp inequality \[ \widehat{\operatorname{vol}}(x,X,\Delta)\le n^{n-1}\operatorname{mld}_x(X,\Delta), \] with equality if and only if $\Delta=0$ near $x$, and analytically, $(x\in X)\cong\frac{1}{r}(1,\ldots,1)$ for some $r\ge1$. We also prove that, in fixed dimension and with coefficients in a fixed finite set, $\widehat{\operatorname{vol}}/\operatorname{mld}$ is discrete away from zero. As applications of the sharp inequality, we obtain lower bounds for minimal log discrepancies of log Fano pairs.

Isogeny classes of abelian varieties over finite fields were described by Tate and Honda. Deligne proved that the category of ordinary abelian varieties over a finite field is equivalent to the category of ordinary Deligne modules. Centeleghe and Stix extended Deligne's results to the whole category of abelian varieties, but, if the base finite field is not prime, then the target category is not related to Deligne modules. In this paper we introduce generalized and twisted Deligne modules and give a more direct generalization of the Deligne theorem.

We give a geometric description of nilpotent representations of equioriented cyclic quivers over a field via an explicit bijection. Over a finite field, this yields formulas for the number of nilpotent representations and for the probability that a representation is nilpotent. We also give a refinement by rank. Over the Boolean semiring, we identify nilpotent semirepresentations with directed acyclic graphs, derive a recursive formula for their number, and determine the asymptotic decay rate of the probability of nilpotence.

Let $G$ be the affine Kac–Moody group of type $\widetilde A_{n-1}$, $B$ be an Iwahori subgroup in $G$, $\mathcal{F}=G/B$ be the flag variety, and $W$ be the Weyl group of $G$. Given distinct involutions $w_1$, $w_2\in W$, we prove that the tangent cones $C_{w_1}$, $C_{w_2}$ to the corresponding Schubert subvarieties $X_{w_1}$ and $X_{w_2}$ of $\mathcal{F}$ at the point $p=e\mod B$ do not coincide as subvarieties of the tangent space to $\mathcal{F}$ at the point $p$. This generalizes similar results in the finite-dimensional setting. The main technical tools we used are combinatorics of the embeddings of the Weyl groups of different ranks and coadjoint orbits for the unipotent radical of the group $B$.

Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $\nu(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(\tau)$ obtained from $F_n$ has a pole of order exactly $\nu(n)$ along the product locus, that $\chi_{10}^{\nu(n)} F_n(\tau)$ is a constant multiple of $G_{n^2}$, and that $\deg_w F_n = k(H_{n^2}) - 10\nu(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.

Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construct, for any $g \geq 3$, a closed subvariety of dimension $g-1$ which contains an open dense subset $\mathcal{U}$ of $a$-invariant $g-2$ such that $\mathcal{U}$ meets every irreducible component of $\mathcal{S}_g$ and every geometric point in $\mathcal{U}$ has automorphism group $\{ \pm 1 \}$ if $p >2$. This gives an independent proof of Oort's conjecture for $p>2$. When $g>3$, so $a = g-2 > 1$, this provides complementary information to the $a=1$ locus investigated in Viehmann's work on how the automorphism groups interact with the geometry.

We explore the interconnections between the monodromy group of stratified bundles on a smooth projective variety $X$ and the monodromy of the strongly semistable vector bundles $V$ on $X$ such that $c_1(V)$ and $c_2(V)$ are numerically trivial.

We develop a theory of singularities of rational maps, focusing on maps $$ f\colon X\dashrightarrow \mathbb P^n, $$ and using their polarised graphs and normalised polarised graphs even when the source $X$ is very singular. To measure singularities of the map at a point $x\in X$, i.e. how far it is from being regular, we introduce invariants including the normalised graph fibre degree $\delta_x(f)$, a generalised lc threshold $\lambda_x(f)$, and invariants measuring the singularities of the normalised graph itself. Numerous examples show that the resulting invariants measure genuinely different aspects of map singularities. We investigate the surface case in detail. We relate the normalised graph fibre degree to multiplicity, prove a sharp threshold–degree inequality for klt surface germs, and develop a detailed theory of linear type maps on smooth and singular surfaces. In particular, we connect the existence of linear type maps to existence of smooth curves through the given point, and with local class groups and complement theory. We conclude with questions and future directions concerning higher dimensions, complements and boundedness, moduli, Cremona groups, commutative algebra, curve-counting theories, and positive characteristic.

For $\gcd(a,b)=1$, we show that the moduli space of $m$-codimensional $\Bbbk[\![T^a,T^b]\!]$-submodules of $\Bbbk[\![T]\!]^n$ is paved by affine cells, by proving that each Białynicki-Birula stratum of a closed related moduli space with respect to the natural $\mathbb{G}_m$-action is an affine bundle over the fixed point locus and that the fixed point locus is an iterated Grassmannian bundle. As an application, we determine the motive of this moduli space in the Grothendieck ring of varieties in terms of an explicit two-variable series $N_{a,b;n}(q,t)$, and use it to explicit compute the groupoid volume of the category of finite modules over $\mathbb{F}_q[\![T^a,T^b]\!]$. The series $N_{a,b;n}$ carries the conjectures we then formulate. At $n=\infty$ we conjecture a bi-infinite family of Rogers–Ramanujan type identities by specializing the $t$-variable; we identify their product side with the normalized character of a module over the $\mathcal{W}$-algebra minimal model $\mathcal{W}_a(a,a+b)$, and observe a connetion to colored Jones tails. At $n<\infty$ we conjecture that $N_{a,b;n}$ is computed by the bottom $\alpha$-row of the trigraded $S^n$-colored HOMFLY homology of the torus knot $T(a,b)$, and that this same bottom row also computes the Quot schemes of finite codimensional $\Bbbk[\![T^a,T^b]\!]$-submoudles of $\Bbbk[\![T^a,T^b]\!]^n$ and the punctual Hilbert schemes of the non-reduced curve $(Y^a-X^b)^n=0$; the three quantities are special values at three points of the trigrading, and when $n=1$ they recover both the conjectures of Oblomkov–Rasmussen–Shende and of Kivinen–Trinh. Finally we conjecture that the one direction of the trigrading these three points do not see is a perverse filtration on the moduli spaces themselves, and we verify its prediction for a smooth germ at $n=2$ by computing the decomposition theorem for the $\mathrm{GL}_2$ spectral-curve family.

We give two related but different methods for constructing rings $R$ of Krull dimension 1 and $K_{-d}(R) \neq 0$. The first works for $d = 2$, and the second works for any $d \geq 2$.

We establish Chudnovsky's Conjecture and, more generally, Demailly's Conjecture for any finite set of points in $\mathbb{P}^n_k$, where $k$ is an algebraically closed field of characteristic 0. The key idea of passing to positive characteristic, as well as the broad plan of the proof, was suggested by ChatGPT-5.6 Sol. All mathematical arguments and proofs in this paper were developed, written, and verified by the authors.

Recently, C. Li constructed stability conditions on the derived categories of all smooth complex projective varieties. These stability conditions satisfy the support property of Kontsevich-Soibelman with respect to the lattice in cohomology generated by powers of an ample class. We extend Li's results to construct stability conditions with full support on Lefschetz type varieties whose algebraic cohomology is generated by divisor classes. This includes all smooth complex projective varieties of dimension at most 3 and smooth projective toric varieties.

The symmetric space $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) ${\rm SL}(n,\Rb)$-invariant metric $d_{X_n}$. In 1939 Teichmüller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\geq 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the total expansion $\TE(f)\in [0,\infty]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of “$k$-dilatation” developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:\Tc_0\to\Tc_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\TE(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the Hilbert-Schmidt expansion $\HE(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.

We construct a six-functor formalism for syntomic cohomology of p-adic formal schemes. This, in particular, generalizes Poincaré duality to general smooth morphisms.

Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $\rho:\pi_1(S_g)\to\PU(2,1)$, let $E_\rho$ be the energy function on Teichmüller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ \rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1) \] with \[ \tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.

Recent advances have revealed that scattering amplitudes in certain quantum field theories admit a geometric formulation in terms of positive geometries. In this framework, amplitudes are encoded by differential forms whose boundary structure reflects physical principles such as locality, unitarity, and factorization. This thesis gives a self-contained introduction to positive geometries, with particular emphasis on the Amplituhedron, which describes amplitudes in planar maximally supersymmetric Yang–Mills theory through its canonical form. We develop three related directions connecting positivity, volumes, and singularities. First, we study positivity properties of canonical forms through dual volume representations. For polytopes, canonical functions compute volumes of dual polytopes; we extend this picture to nonlinear positive geometries, uncovering non-negative transcendental measures and links with complete monotonicity. Second, we investigate loop-level amplitudes via their singularity structure. Combining Amplituhedron geometry with Landau analysis, we classify leading singularities of the Wilson loop with Lagrangian insertion and constrain the possible singular loci. Third, we examine conjectural relations between Landau singularities, positivity, and cluster algebras. Using momentum-twistor and Grassmannian methods, we identify recursive structures across loop orders, prove several infinite families of cases, and propose a strategy toward the general conjectures. Overall, the thesis develops the perspective that positive geometry provides a unifying language for the analytic and geometric organization of scattering amplitudes, from canonical forms and volumes to the singularities that emerge after loop integration.