We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.
The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of Q-Divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in P^r is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree d plane curves is explicitly evaluated.
The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the $\ell$-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The $\ell$-adic cohomology groups and their decomposition into eigenspaces are s…
The $P = W$ conjecture identifies the perverse filtration of the Hitchin system on the cohomology of the moduli space of Higgs bundles with the weight filtration of the corresponding character variety. In this paper, we introduce an enumerative approach to to this problem; our technique only uses the structure of the equivariant intersection numbers on the moduli space of Higgs bundles, and little information about the topology of the Hitchin map. In the rank $2$ case, starting from the known intersection numbers of the moduli of stable bundles, we derive the equivariant intersection numbers o…
For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.
The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos.
We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated to a family of stable curves, we construct a bundle map ('degree-2 Brill-Noether') from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton-Macpherson residual intersection theory, th…
These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel's A1-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A1-Milnor number inspired by the first named author's enrichment of dynamic intersection numbers.
ORIGINS Data Science Lab（ORIGINS数据科学实验室）；Technical University Munich（慕尼黑技术大学）；Department of Mathematics（数学系）；Harvard University（哈佛大学）；Departement Mathematik（数学系）；ETH Zürich（苏黎世联邦理工学院）
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
We investigate generalizations along the lines of the Mordell--Lang conjecture of the author's $p$-adic formal Manin--Mumford results for $n$-dimensional $p$-divisible formal groups $\mathcal{F}$. In particular, given a finitely generated subgroup $Γ$ of $\mathcal{F}(\overline{\mathbb{Q}}_p)$ and a closed subscheme $X\hookrightarrow \mathcal{F}$, we show under suitable assumptions that for any points $P\in X(\mathbb{C}_p)$ satisfying $nP\inΓ$ for some $n\in\mathbb{N}$, the minimal such orders $n$ are uniformly bounded whenever $X$ does not contain a formal subgroup translate of positive dimens…
We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons.
We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. It is possible to prove in this theory analogues of the Adjunction formula and the Riemann-Roch formula.
We study enumerative questions on the moduli space $\mathcal{M}(L)$ of hyperplane arrangements with a given intersection lattice $L$. Mnëv's universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it is even difficult to compute the dimension $D =\dim \mathcal{M}(L)$. Embedding $\mathcal{M}(L)$ in a product of projective spaces, we study the degree $N=\mathrm{deg} \mathcal{M}(L)$, which can be interpreted as the number of arrangements in $\mathcal{M}(L)$ that pass through $D$ points in general position. For generic arrangements $N$ can be computed combinat…
We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.
We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterlé) and to the study of random $p$-adic polynomial systems of equations.
Given six points $A,B,C,D,E,F$ on a nonsingular conic in the complex projective plane, Pascal's theorem says that the three intersection points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called a pascal, and we get altogether $60$ such lines by permuting the points. In this paper, we consider the enumerative problem of finding the number of sextuples $(A, B, \dots, F)$ which correspond to three pre-specified pascals. We use computational techniques in commutative algebra to solve this problem in all cases. The results are tabulated using the so-called 'dual…
We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.
We establish a criterion for sheaves on an adically complete DG scheme to be coherent. We deduce a description of coherent sheaves on an adically complete lci singularity in terms of modules for a DG Lie algebra.
The first part of this work constructs real positive-genus Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants. The present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former. After describing large collections of real-orientable symplectic manifolds, we show that the real genus 1 Gromov-Witten invariants of sufficiently positive almost Kahler threefolds are s…
We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these numbers are useful for computing genus-one enumerative invariants of projective spaces and Gromov-Witten invariants of complete intersections.
This paper has been rendered obsolete by our newer eprint alg-geom/9411005 "Bott's formula and enumerative geometry", which is a considerably expanded version of the same paper, in spite of the change of titles. Please download alg-geom/9411005 instead, and update possible references.
We study the relative Hilbert scheme of a family of nodal (or smooth) curves via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We derive an intersection calculus for Chern classes of tautological bundles on the relative Hilbert scheme, which has applications to enumerative geometry.
We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a semipositive symplectic manifold of dimension 4, when GW([point],...,[point]) is enumerative. In particular, we show that the intersection theoretic property depends on the existence of a critical point on a stable map.
The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough w.r.t. the degree. As a reward we get a number of surprising enumerative formulas relating lines, conics and twisted cubics on X.
This is an expanded version of the author's talk at the third annual meeting of International Consortium of Chinese Mathematicians held at USTC in December 2020. In this expository article, we give a survey on joint works with Hosono, Lian, and Yau in arXiv:2003.07148 and arXiv:2008.04039. We also carry out some explicit examples to illustrate our results in enumerative geometry which will appear in our forthcoming papers.
Slopes and weights of $\ell$-adic cohomology of rigid spaces
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of the Chinese Academy of Sciences（中国科学院大学）
We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We work out the action of the blowup or 'discriminant' polarization on some natural cycles in the Hilbert scheme, including generalized diagonals and cycles, called 'node scrolls', parametrizing schemes supported on singular points. We derive an intersection calculus for Chern classes of tautologic…
A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in $\mathbb{P}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbol…
This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $…
Given a countable residually finite group $Γ$, we write $Γ_n \to e$ if $(Γ_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $Γ_n$'s contains only the unit element $e$ of $Γ$. Given a $Γ$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} χ(Γ_n , M) = \prod_i |H_i (Γ_n , M)|^{(-1)^i} \end{equation} and the limit in the field $\mathbb{Q}_p$ of $p$-adic numbers \begin{equation} h_p := \lim_{n\to\infty} (Γ: Γ_n)^{-1} \log_p χ(Γ_n , M) \; . \end{equation} Here $\log_p : \mathbb{Q}^{\times}_p \to \mathbb{Z}_p$ is the …
We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, o…
This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree $d$ by doing intersection theory on moduli spaces.
We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $Ω$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $Ω$-classes by…
The Chabauty--Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental $p$-adic analytic functions arising from certain Selmer schemes associated to the unipotent fundamental group of the variety. In this paper we establish several foundational results on the Chabauty--Kim method for curves over number fields. The two main ingredients in the proof of these results are an unlikely intersection result for zeroes of iterated integrals, and a careful analysis of the intersection of the Selmer scheme of the original curve with the u…
We provide formulas for the degrees of the projections of the locus of square matrices with given rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations stem from applications to `matrix rigidity'; we also view them as an excellent source of examples to test methods in intersection theory, particularly computations of Segre classes. Our results are generally expressed in terms of intersection numbers in Grassmannians, which can be computed explicitly in many cases. We observe that, surprisingly (to us), these degrees appear to match the numbers of …
For a nonsingular projective variety $X$, the virtual Tevelev degree in Gromov-Witten theory is defined as the virtual degree of the morphism from $M_{g,n}(X,d)$ to the product $M_{g,n} \times X^n$. After proving a simple formula for the virtual Tevelev degree in the (small) quantum cohomology ring of $X$ using the quantum Euler class, we provide several exact calculations for flag varieties and complete intersections. In the cominuscule case (including Grassmannians, Lagrangian Grassmannians, and maximal orthogonal Grassmannians), the virtual Tevelev degrees are calculated in terms of the eig…
We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N \geq 2$, and consider an isolated complete intersection curve singularity germ $f \colon (\mathbb{C}^N,0) \to (\mathbb{C}^{N-1},0)$. We introduce a numerical function $m \mapsto \operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{\mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, \dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $\operato…
In this paper, we study special cycles on the Krämer model of $\mathrm{U}(1,1)(F/F_0)$-Rapoport-Zink spaces where $F/F_0$ is a ramified quadratic extension of $p$-adic number fields with the assumption that the $2$-dimensional hermitian space of special quasi-homomorphisms is anisotropic. We write down the decomposition of these special cycles and prove a version of Kudla-Rapoport conjecture in this case. We then apply the local results to compute the intersection numbers of special cycles on unitary Shimura curves and relate these intersection numbers to Fourier coefficients of central deriva…
In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. N. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher cod…
We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne'…
We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue field $k$ and interpret the conjecture in this case as a remarkable identity between the number of $k$-rational points of them. We prove the corresponding identity for the numbers of $k_f$-rational points, for any extension of even degree $f$ of $k$. The proof uses the local intersection theory on a regular surface and Deligne's theory of intersection multiplic…
We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_q$. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the $\ell$-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree $n$ in the limit as $q \to \infty$. The results of this paper can be v…
