Intersection Multiplicities Research Articles

A vanishing theorem for oriented intersection multiplicities

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for oriented intersection multiplicities of Serre's vanishing result for intersection multiplicities. It also suggests a Vanishing Conjecture for arbitrary regular local rings containing 1/2, which is analogous to Serre's (which was proved independently by Roberts, and Gillet and Soule).

Intersection Numbers of Heegner Divisors on Shimura Curves

In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X0(N) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. So we certainly expect that all of [GKZ] should generalize similarly. Our work here can be viewed as a step in this direction. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was done.

Intersection multiplicity of Serre on regular schemes

The study of the intersection multiplicity function χ O X ( F , G ) over a regular scheme X for a pair of coherent O X -modules F and G is the main focus of this paper. We mostly concentrate on projective schemes, vector bundles over projective schemes, regular local rings and their blow-ups at the closed point. We prove that (a) vanishing holds in all the above cases, (b) positivity holds over Proj of a graded ring finitely generated over its 0th component which is artinian local, when one of F and G has a finite resolution by direct sum of copies of O ( t ) for various t, and (c) non-negativity holds over P R n , R regular local, and over arbitrary smooth projective varieties if their tangent bundles are generated by global sections. We establish a local–global relation for χ for a pair of modules over a regular local ring via χ of their corresponding tangent cones and χ of their corresponding blow-ups. A new proof of vanishing and a special case of positivity for Serre's Conjecture are also derived via this approach. We also demonstrate that the study of non-negativity is much more complicated over blow-ups, particularly in the mixed characteristics.

Antispecial cycles on the Drinfeld upper half plane and degenerate Hirzebruch–Zagier cycles

We define the notion of antispecial cycles on the Drinfeld upper half plane in analogy to the notion of special cycles in Kudla and Rapoport (Invent Math 142:153–223, 2000). We determine equations for antispecial cycles and calculate the intersection multiplicity of two antispecial cycles. The result is applied to calculate the intersection multiplicity of certain degenerate Hirzebruch–Zagier cycles. Finally we compare this intersection multiplicity to certain representation densities.

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov–Witten theory the numbers of complex plane rational curves of degree d through 3 d − 1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.

On the number of subrepresentations of a general quiver representation

AbstractIt is well known that the intersection multiplicities of Schubert classes in the Grassmannian are Littlewood-Richardson coefficients. We generalize this statement in the context of quiver representations. Here the intersection multiplicity of Schubert classes is replaced by the number of subrepresentations of a general quiver representation, and the Littlewood-Richardson coefficients are replaced by the dimension of a certain space of semi-invariants.

On generalized Kneser hypergraph colorings

In 2002, the second author presented a lower bound for the chromatic numbers of hypergraphs KG s r S , “generalized r-uniform Kneser hypergraphs with intersection multiplicities s .” It generalized previous lower bounds by Kříž (1992/2000) for the case s = ( 1 , … , 1 ) without intersection multiplicities, and by Sarkaria (1990) for S = ( [ n ] k ) . Here we discuss subtleties and difficulties that arise for intersection multiplicities s i > 1 : (1) In the presence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,” depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version. (2) The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r. Currently we have no valid proof for the lower bound result in the other cases. We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.

On Fermat curves and maximal nodal curves

Let f(x) = a(x − α1) · · · (x − αn), a > 0, be a real polynomial with n distinct real roots; it has [(n − 1)/2] maxima and (n − 1) − [(n − 1)/2] minima. Thom has studied the space of real polynomials and showed, for example, that any given polynomial f can be deformed into a special polynomial that has the same maxima and minima [8]. A typical such polynomial is the Chebyshev polynomial. A nodal curve C is an irreducible plane curve of degree n that contains only nodes (= A1 singularities). A nodal curve is called a maximal nodal curve if it is rational and nodal; by Plucker’s formula, it must contain (n−1)(n−2) 2 nodes to be maximal. In the space of polynomials of two variables, a maximal nodal curve can be understood as a generalization of a Chebyshev polynomial. In [6] the author constructed a maximal nodal curve of join type f(x)+ g(y) = 0 using a Chebyshev polynomial f(x) and a similar polynomial g(y) that has one maximal value and two minimal values. In this paper we present another extremely simple way, for a given integern > 2, to construct a beautifully symmetric and maximal nodal curve Dn as a by-product of the geometry of the Fermat curve xn+1 + y n+1 + 1 = 0. A smooth point P of a plane curve C is called a flex point of flex-order k ≥ 3 if the tangent line TP at P and C intersect with intersection multiplicity k. The maximal nodal curve Dn, which we construct in this paper, contains three flexes of flex-order n, and it is symmetric with respect to the permutation of three variables U,V,W. By a special case of Zariski [10] and Fulton [2], π1(P − C) = Z/nZ if C is a maximal nodal curve of degree n. The examples Dn provide an alternate proof. Zariski [10] observed that the fundamental group of the complement of an irreducible curve C of degree n is abelian if C has a flex of flex-order either n or n− 1. Since the moduli of maximal nodal curves of degree n is irreducible (by Harris [3]), the claim follows. For the construction, we start from the Fermat curve Fn : x + y n +1 = 0 and study singularities of the dual curve Fn. The Fermat curve and the dual curve Fn have canonical Z/nZ×Z/nZ actions, so the defining polynomial of Fn is written as h(u, v) = 0 for a polynomial h(u, v) of degree n− 1. The curve h(u, v) = 0 defines our maximal nodal curve Dn−1. Geometrically this is the quotient of the

Constructing modules of finite projective dimension with prescribed intersection multiplicities

Recently Roberts and Srinivas proved the existence of large classes of modules of finite length and finite projective dimension with prescribed intersection multiplicities, giving many new counterexamples to earlier vanishing conjectures. We present an explicit method for constructing the examples whose existence is implied by their theorem.

The upper bound of Frobenius related length functions

In this paper, we study the asymptotic behavior of lengths of Tor modules of homologies of complexes under the iterations of the Frobenius functor in positive characteristic. We first give upper bounds to this type of length functions in lower dimensional cases and then construct a counterexample to the general situation. The motivation of studying such length functions arose initially from an asymptotic length criterion given in [S.P. Dutta, Intersection multiplicity of modules in the positive characteristics, J. Algebra 280 (2004) 394–411] which is a sufficient condition to a special case of nonnegativity of χ ∞ . We also provide an example to show that this sufficient condition does not hold in general.

On the upper semi-continuity of the Hilbert–Kunz multiplicity

We show that the Hilbert–Kunz multiplicity of a d-dimensional non-regular complete intersection over F p ¯ , p > 2 prime, is bounded by below by the Hilbert–Kunz multiplicity of ∑ i = 0 d x i 2 = 0 , answering positively a conjecture of Watanabe and Yoshida in the case of complete intersections.

Algorithm for the Semigroup of a Space Curve Singularity

The semigroup of values of irreducible space curve singularities is the set of intersection multiplicities among hypersurfaces and the given curve. It is an invariant of the singularity, and for plane curves it characterizes the equisingularity type considered by Zariski. For space curve singularities the semigroup of values is a numerical semigroup and it can not be computed by means of the exponents of any Puiseux parametrization, as in the plane case. We obtain an algorithm for calculating the semigroup of values of a space curve singularity, which determines the generators of the semigroup and the valuation ideals associated with the semigroup. We give a Maple version of the algorithm.

Degree formulae for offset curves

In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S , that is defined depending on a non-empty Zariski open subset of R 2 . The second formula is based on the resultant of the defining polynomial of C , and the polynomial defining generically S . The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C , at their intersection points.

Intersection multiplicity of modules in the positive characteristics

We introduce an asymptotic length criteria on Tor 0 and Tor 1 of a pair of modules, via the Frobenius map, to study positivity and non-negativity of Serre-intersection multiplicity over local complete intersections and Gorenstein rings in the positive characteristics, when both modules have finite projective dimension. We use this criteria to derive a sufficient condition for (a) non-negativity over complete intersections when one of the modules has dimension ⩽2, and (b) non-negativity over Gorenstein rings of dimension ⩽5. We also show that the problem of intersection multiplicity on local rings (complete intersection, Gorenstein, Cohen–Macaulay) in characteristic 0 can be reduced to local rings (complete intersection, Gorenstein, Cohen–Macaulay respectively) in the positive characteristics.

Relative morsification theory

In this paper we develop a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known morsification results for non-isolated singularities and generalise them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalisations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalised jet-space we show how to interpret the newly defined invariants as certain intersection multiplicities; finally, we characterise which invariants can be interpreted as intersection multiplicities in the above mentioned generalised jet space.

Numerical equivalence defined on Chow groups of Noetherian local rings

In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb Q}$-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck's standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen-Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert-Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.

Bounds for curves in abelian varieties

A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's differential algebra. The problem is modeled after the abc-Conjecture of Masser-Oesterle for abelian varieties over the function field of a curve. As an application a finiteness theorem is proved for maps from a curve into an abelian variety omitting an ample divisor.

Relative morsification theory

In this paper we develop a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known morsification results for non-isolated singularities and generalise them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalisations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalised jet-space we show how to interpret the newly defined invariants as certain intersection multiplicities; finally, we characterise which invariants can be interpreted as intersection multiplicities in the above mentioned generalised jet space.

Serre's vanishing conjecture for Ext-groups

Let R be a noetherian commutative local ring, and M, N be finitely generated R-modules. Then a generalized form of Serre's Vanishing conjecture can be stated as follows: if (1) length ( M⊗ R N)<∞, (2) pd( M),pd( N)<∞, and (3) dim M+ dim N< dim R , then χ(M,N)≔ ∑ i=0 ∞ (−1) i length( Tor i R(M,N))=0. It is known that Serre's Vanishing conjecture holds for a complete intersection ring R, but is not known for a Gorenstein ring R. We can make a similar conjecture replacing Tor by Ext, namely, if M and N satisfy the above three conditions, then ξ(M,N)≔ ∑ i=0 ∞ (−1) i length( Ext R i(M,N))=0. In this paper, we will prove that, over a Gorenstein ring R, the Tor-version of the Vanishing conjecture, the Ext-version of the Vanishing conjecture, and the commutativity of the intersection multiplicity defined in Mori and Smith (J. Pure Appl. Algebra 157 (2001) 279) are all equivalent. Further, we will prove that a certain Ext-version of the Vanishing conjecture holds for a large class of noncommutative projective schemes, typically including all commutative projective schemes over a field, by extending Bézout's Theorem.

Approximation by smooth curves near the tangent cone

We show that through a point of an affine variety there always exists a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This result has an application to the study of affine schemes.