Intersection Multiplicities Research Articles

Positivity of intersection multiplicity over a two-dimensional base

We prove the positivity of Serre's Intersection Multiplicity for regular local rings that are essentially smooth over a two-dimensional, regular base. Afterward, we apply this result to prove a transversality theorem for unramified regular local rings via a local analysis on the blowup.

Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for virtually polycyclic subgroups and in particular finitely generated virtually nilpotent groups of local biholomorphisms. We provide several methods to identify this property and build examples. As a consequence we generalize results of Arnold, Seigal-Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups.

On Drinfeld modular forms of higher rank III: The analogue of the k/12-formula

Continuing the work of [8] and [9], we derive an analogue of the classical “k/12-formula” for Drinfeld modular forms of rank r≥2. Here the vanishing order νω(f) of one modular form at some point ω of the complex upper half-plane is replaced by the intersection multiplicity νω(f1,…,fr−1) of r−1 independent Drinfeld modular forms at some point ω of the Drinfeld symmetric space Ωr. We apply the formula to determine the common zeroes of r−1 consecutive Eisenstein series Eqi−1, where n−r<i<n for some n≥r.

Faltings heights of abelian varieties with complex multiplication

Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions. As an application of this result, we prove an averaged version of Colmez’s conjecture on the Faltings heights of CM abelian varieties.

Elliptic surfaces and contact conics for a 3-nodal quartic

Let ${\mathcal Q}$ be an irreducible $3$-nodal quartic and let ${\mathcal C}$ be a smooth conic such that ${\mathcal C} \cap {\mathcal Q}$ does not contain any node of ${\mathcal Q}$ and the intersection multiplicity at $z \in {\mathcal C} \cap {\mathcal Q}$ is even for each $z$. In this paper, we study geometry of ${\mathcal C} + {\mathcal Q}$ through that of integral sections of a rational elliptic surface which canonically arises from ${\mathcal Q}$ and $z \in {\mathcal C} \cap {\mathcal Q}$. As an application, we construct Zariski pairs $({\mathcal C}_1 + {\mathcal Q}, {\mathcal C}_2 + {\mathcal Q})$, where ${\mathcal C}_i$ $(i = 1, 2)$ are smooth conics tangent to ${\mathcal Q}$ at four distinct points.

Divergent series and Serre's intersection formula for graded rings

On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the divergent series from Serre's formula can be related to these fractional multiplicities. We apply work of Avramov and Buchweitz to answer Fulton's question in the context of graded rings.

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

PLANE CURVES MEETING AT A POINT WITH HIGH INTERSECTION MULTIPLICITY

As a generalization of an inflection point, we consider a point P on a smooth plane curve C of degree m at which another curve C' of degree n meets C with high intersection multiplicity. Especially, we deal with the existence of two curves of degree m and n meeting at the unique point.

The log-canonical threshold of a plane curve

AbstractWe give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27].

FINITENESS PROPERTIES OF FORMAL LIE GROUP ACTIONS

Following the ideas of Arnold and Seigal–Yakovenko, we prove that the space of matrix coefficients of a formal Lie group action belongs to a Noetherian ring. Using this result, we extend the uniform intersection multiplicity estimates of these authors from the abelian case to general Lie groups. We also demonstrate a simple new proof for a jet-determination result of Baouendi et al.

Pfaffian intersections and multiplicity cycles

We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the variety. We show that this multiplicity can be majorized at every point p by the local algebraic multiplicity at p of a suitably constructed algebraic cycle. The construction is based on Gabrielov’s complex analog of the Rolle–Khovanskiĭ lemma. We illustrate the main result by deriving similar uniform estimates for the complexity of the Milnor fiber of a deformation (under a smoothness assumption) and for the order of contact between an algebraic hypersurface and an arbitrary non-singular one-dimensional foliation. We also use the main result to give an alternative geometric proof for a classical multiplicity estimate in the context of commutative group varieties.

Doing algebraic geometry with the RegularChains library

Traditionally, Groebner bases and cylindrical algebraic decomposition are the fundamental tools of computational algebraic geometry. Recent progress in the theory of regular chains has exhibited efficient algorithms for doing local analysis on algebraic varieties. In this note, we present the implementation of these new ideas within the module AlgebraicGeometryTools of the RegularChains library. The functionalities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of calculation has several applications like computing the Zarisky closure of a constructible set as well as computing tangent cones of space curves, thus providing an alternative to the standard approaches based on Groebner bases and standard bases, respectively. From there, we have derived an algorithm which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points. This algorithm relies only on the manipulations of regular chains.

Secondary fans and tropical Severi varieties

This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Severi variety cannot be a subfan of the corresponding secondary fan. The second theorem provides a partial converse, namely, we find conditions under which a cone of the secondary fan is fully contained in the tropical Severi variety. As a first application of these results, we also find a combinatorial formula for the tropical intersection multiplicities for secondary fans.

Intersection multiplicity of Serre in the unramified case

We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is unramified, then just as in the equicharacteristic case, the intersection multiplicity of two modules is bounded below by the product of their Hilbert–Samuel multiplicities. We also explain, in terms of the blowup of SpecA, the geometric significance of achieving this lower bound.

The grade conjecture and asymptotic intersection multiplicity

Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\pd M < \infty$, we study the asymptotic intersection multiplicity $\chi_\infty(M, A/\underline{x})$, where $\underline{x} = (x_1, \ldots, x_r)$ is a system of parameters for $M$. We show that there exists a system of parameters such that $\chi_\infty$ is positive if and only if $\dim \Ext^{d-r}(M, A) = r$, where $d = \dim A$ and $r = \dim M$. We use this to prove several results relating to the Grade Conjecture, which states that $\grade M + \dim M = \dim A$ for any module $M$ with $\pd M < \infty$.

Hochster's theta pairing and numerical equivalence

AbstractLet (A, ) be a local hypersurface with an isolated singularity. We show that Hochster's theta pairing θA vanishes on elements that are numerically equivalent to zero in the Grothendieck group of A under the mild assumption that Spec A admits a resolution of singularities. This extends a result by Celikbas-Walker. We also prove that when dimA = 3, Hochster's theta pairing is positive semi-definite. These results combine to show that the counter-example of Dutta-Hochster-McLaughlin to the general vanishing of Serre's intersection multiplicity exists for any three dimensional isolated hypersurface singularity that is not a UFD and has a desingularization. We also show that, if A is three dimensional isolated hypersurface singularity that has a desingularization, the divisor class group is finitely generated torsion-free. Our method involves showing that θA gives a bivariant class for the morphism Spec (A/) → Spec A.

Local dynamics of intersections: V. I. Arnold’s theorem revisited

V. I. Arnold proved in 1993 that the intersection multiplicity between two germs of analytic subvarieties at a fixed point of a holomorphic invertible self-map remains bounded when one of the germs is dragged by iterations of the self-map. The proof is based on the Skolem-Mahler-Lech theorem on zeros in recurrent sequences.We give a different proof, based on the Noetherianity of certain algebras, which allows one to generalize Arnold’s theorem for local actions of arbitrary finitely generated commutative groups, with both discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails.

On the growth of local intersection multiplicities in holomorphic dynamics: a conjecture of Arnold

We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.

A Degree Formula for Secant Varieties of Curves

AbstractUsing the Stückrad–Vogel self-intersection cycle of an irreducible and reduced curve in pro-jective space, we obtain a formula that relates the degree of the secant variety, the degree and the genus of the curve and the self-intersection numbers, the multiplicities and the number of branches of the curve at its singular points. From this formula we deduce an expression for the difference between the genera of the curve. This result shows that the self-intersection multiplicity of a curve in projectiveN-space at a singular point is a natural generalization of the intersection multiplicity of a plane curve with its generic polar curve. In this approach, the degree of the secant variety (up to a factor 2), the self-intersection numbers and the multiplicities of the singular points are leading coefficients of a bivariate Hilbert polynomial, which can be computed by computer algebra systems.

An example of the Weierstrass semigroup of a pointed curve on K3 surfaces

Let X be a K3 surface with Picard number one which is given by a double cover π:X→ℙ2. Let C be a smooth curve on X with π −1 π(C)=C which is not the ramification divisor of π, and let P be a ramification point of π| C :C→π(C). In this paper, in the case where the intersection multiplicity at π(P) of the curve π(C) and the tangent line at π(P) on π(C) is equal to deg(π(C)) or deg(π(C))−1, we investigate the Weierstrass semigroup of the pointed curve (C,P).