Enumerative Geometry Research Articles

The enumerative geometry of projective algebraic surfaces and the complexity of aspect graphs

The aspect graph is a popular viewer-centered representation that enumerates all the topologically distinct views of an object. Building the aspect graph requires partitioning viewpoint space in view-equivalent cells by a certain number of visual event surfaces. If the object is piecewise-smooth algebraic, then all visual event surfaces are either made of lines having specified contacts with the object or made of lines supporting the points of contacts of planes having specified contacts with the object. In this paper, we present a general framework for studying the enumerative properties of line and plane systems. The context is that of enumerative geometry and intersection theory. In particular, we give exact results for the degrees of all visual event surfaces coming up in the construction of aspect graphs of piecewise-smooth algebraic bodies. We conclude by giving a bound on the number of topologically distinet views of such objects.

Piecewise polynomial functions, convex polytopes and enumerative geometry

0. Introduction. This paper explores some of the connections between the objects of its title. It is based on a new approach to McMullen’s polytope algebra, and on its relation with equivariant cohomology of toric varieties. In particular, we give another proof of a recent result of Fulton and Sturmfels, which identifies the polytope algebra with the direct limit of all Chow rings of smooth, complete torus embeddings (see [14]). On the other hand, we obtain a version of the classical theorem of Bezout, which holds in any spherical homogeneous space. This generalizes a theorem of Bernstein and Kouchnirenko: The number of common points to d hypersurfaces in general position in a d-dimensional torus is d! times the mixed volume of the associated Newton polytopes (see [2], [18] and also [13] 5.5). Given a finite-dimensional vector space V over an ordered field K, there is a wellknown correspondence between convex polytopes in the dual space V ∗ and piecewise linear convex functions on V . Namely, to any convex polytope, we associate its support function. Denote by R the algebra generated by the support functions of all convex polytopes, in the algebra of continuous functions on V . In the first section of this paper, we study the algebra R when K is the field of rational numbers. It turns out (see 1.5) that R is the algebra of continuous, piecewise polynomial functions on V ; in particular, R contains the algebra of polynomial functions. We prove that any choice of coordinate functions on V defines a regular sequence in R; moreover, the quotient of R by the ideal generated by V ∗ is isomorphic to the rational polytope algebra of McMullen (see 1.3, 1.5; our proof is based on work of Morelli, see [21]). This explains the non-trivial grading of the polytope algebra, by the obvious grading of R. More generally, the quotients of R by powers of the ideal generated by V ∗, are isomorphic to the higher versions of the polytope algebra, considered recently by McMullen, Pukhlikov and Khovanskii; see [20], [24]. In fact, we study the algebra R as the direct limit of its subalgebras RΣ consisting of functions which are piecewise polynomial with respect to a fixed fan Σ. To such a fan is

On adelic Chern forms and the Bott residue formula

0. Introduction. The Bott Residue Formula gained renewed attention recently due to its use in enumerative algebraic geometry (cf. [ES], [Ko]). If X is a smooth projective variety over a field k of characteristic 0, then Bott’s formula makes sense purely algebraically, with the Chern classes taken in the algebraic De Rham cohomology H·DR(X/k). In this paper we survey an algebraic proof of the formula using Beilinson adeles, which was discovered by R. Hubl and the author (see [HY]). Suppose v ∈ Γ(X, TX) is a global vector field with isolated, simple, k-rational zeroes (see Remark 3.2 for generalizations). Let E1, . . . , Em be locally free OX -modules. Suppose Λi is an action of v on Ei, i.e. a differential operator Λi : Ei → Ei satisfying Λi(ae) = v(a)e + aΛi(e) for local sections a ∈ OX , e ∈ Ei. Suppose Q(ti,j) is a homogeneous polynomial of degree n = dimX in the variables ti,j (i = 1, . . . ,m; j = 1, . . . , ri; ri := rank Ei) which have degrees deg ti,j = j. For a zero z of v let us denote by Λi|z the restriction of Λi to Ei|z := Ei ⊗ k(z), which is a k-linear endomorphism. Also let us denote by ad v|z the restriction of ad v to TX ⊗ k(z); this is invertible. We let Pi denote the ith conjugation-invariant polynomial on matrices (of unspecified size). Finally let ∫

Homological geometry I. Projective hypersurfaces

Introduction Consider a generic quintic hypersurfaceX inCP . It is an example of Calabi– Yau 3-folds. It follows from Riemann–Roch formula, that rational curves on a generic Calabi–Yau 3-fold should be situated in a discrete fashion. Therefore a natural question of enumerative algebraic geometry arises: find the number nd of rational curves in X of degree d for each d = 1, 2, 3, .... In 1991 Candelas, de la Ossa, Green and Parkes [1] ‘predicted’ all the numbers nd simultaneously: they conjectured that the generating function

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV 1 ,...,V r in a toric varietyP Σ and the system of differential operators annihilating the special generalized hypergeometric series Φ0 constructed from the fan Σ. Using this generalized hypergeometric series, we propose conjectural mirrorsV′ ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.In the second part of the paper we consider some examples of Calabi-Yau 3-folds having Picard number >1 in products of projective spaces. For conjectural mirrors, using the recurrent relation among coefficients of the restriction of the hypergeometric function Φ0 on a special line in the moduli space, we determine the Picard-Fuchs equation satisfied by periods of this special one-parameter subfamily. This allows to obtain some sequences of integers which can be conjecturally interpreted in terms of Gromov-Witten invariants. Using standard techniques from enumerative geometry, first terms of these sequence of integers are checked to coincide with numbers of rational curves on Calabi-Yau 3-folds.

The Enumerative Geometry of Plane Cubics. I: Smooth Cubics

The Enumerative Geometry of Plane Cubics. I: Smooth Cubics

Enumerative geometry of singular plane curves

general points in the plane? The answer is easy for 6 = 1 (namely 3 (d l ) 2) but otherwise was, to my knowledge, unknown until now. We are going to develop a recursive procedure for solving such problems. To make the recursion possible, it will be necessary to consider a generalization of independent interest of the above problem. Let S be the blow-up of IP 2 at a point P, with exceptional divisor E1 and a fixed line section Eo (we call S an irreducible fan). A curve C c S is said to be of type (d, e) if it is the proper transform of a curve of degree d in ~z having multiplicity e at P. One then asks: how many curves C c S of given type (d, e), having only smooth branches of given configuration along E0 w E1 and otherwise having a given number of nodes as their only singularities, pass through the "appropriate" number of points of S, some of which may be on E o w E~ ? We will solve this problem by recursion on d e . The recursion is based on degenerating the surface S to a surface (called a 2-fan) of the form So = $1 w S 2 with Si--S, i = i, 2. Using and suitably enhancing methods of [3], we can describe a suitable part of the limit on So of the relevant families of curves of type (d, e) on S, d > e + 1, in terms of similar families of curves of type (d,e') and (e', e), e ' < e = d on $1 and $2, respectively.

Enumerative geometry of triangles, II

Enumerative geometry of triangles, II

Enumerative geometry of triangles, I

Enumerative geometry of triangles, I

Enumerative Geometry of Rational Space Curves

Enumerative Geometry of Rational Space Curves

Complete conic of S 2 and their model variety Ω 5 102 [27

This is a study of the aggregate of complete conics in the plane S 2 and their representation in S 27 by the points of a model variety Q of dimension 5 and order 102. Equivalence bases in each dimension on O are obtained and described, together with bases on certain important subvarieties of s2, with a particular view to clarifying the enumerative geometry of complete conics. A number of important and interesting systems of complete conics are studied in some detail.

Counting divisors with prescribed singularities

Given a family of divisors { D s } \{ {D_s}\} in a family of smooth varieties { Y s } \{ {Y_s}\} and a sequence of integers m 1 , … , m t {m_1}, \ldots ,{m_t} , we study the scheme parametrizing the points ( s , y 1 , … , y t ) (s,{y_1}, \ldots ,{y_t}) such that y i {y_i} is a (possibly infinitely near) m i {m_i} -fold point of D s {D_s} . We obtain a general formula which yields, as special cases, the formula of de Jonquières and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.

On the cohomology of real Grassmanians

LetGk(Rn+k){G_k}({\textbf {R}^{n + k}})denote the grassman manifold ofk-planes in real(n+k)(n\, +\, k)-space andw1∈H1(Gk(Rn+k);Z2){w_1}\, \in \, {H^1}({G_k}({\textbf {R}^{n + k}});\,{\textbf {Z}_2})the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height ofw1{w_1}in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.

Coincidence formulas in enumerative geometry

(1979). Coincidence formulas in enumerative geometry. Communications in Algebra: Vol. 7, No. 16, pp. 1685-1711.

On q-th Derivative of Vector Bundles

In the present note we shall be concerned with the improvement of fundamental definitions in higher order enumerative geometry which has been recently given by W. F. Pohl. Pohl’s definition of q-th derivative of vector bundle is very complicated. We shall give a simpler and more reasonable definition of the q-th derivative of vector bundle in terms of sheaf theory and simplify the proofs in [P]. We shall also give a definition of higher order singularity of map.

On the enumerative geometry of triangles

A triangle , in the sense of Schubert [1], is an entity in the plane S 2 consisting of (a) an ordered triad of points ( P 1 , P 2 , P 3 ); (b) an ordered triad of lines ( l 1 , l 2 , l 3 ) connected with the points P i by the incidence relations P i ⊂ l j ( i ≠ j ); and (c) a three base-point net Φof conies with the P i as base-points.

Il passato e il presents delle principali teorie geometriche; storia e bibliografia

THE first edition of Prof. Loria's treatise on the history and bibliography of geometry was an expansion of a course of lectures delivered at Genoa in 1886. After an introductory chapter on geometry up to 1850, there followed eleven more dealing respectively with higher plane curves, surfaces, space curves, differential geometry, analysis situs, line geometry, correspondences and transformations, enumerative geometry, non-Euclidean geometry, hyperspace, and miscellaneous topics. This was translated into German and Polish. The second edition added references up to 1896. The fourth edition is doubled in size, the extra matter forming a second part, on a similar plan to the first, but dealing with contributions since 1896. It is remarkable that so much good work has been done in this period, which includes the War years. The book will be a most valuable aid to geometrical research. Il passato e il presents delle principali teorie geometriche; storia e bibliografia. Prof. Gino Loria Per. Quarta edizione totalmente rifatta. Pp. xxiii + 467. (Padova: Casa Editrice Dott. Antonio Milani, 1931.) 60 lire.

A Point in Enumerative Geometry

Journal of the London Mathematical SocietyVolume s1-1, Issue 3 p. 167-170 Notes and Papers A Point in Enumerative Geometry J. H. Grace, J. H. GraceSearch for more papers by this author J. H. Grace, J. H. GraceSearch for more papers by this author First published: July 1926 https://doi.org/10.1112/jlms/s1-1.3.167AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volumes1-1, Issue3July 1926Pages 167-170 RelatedInformation

Discussions: The "Limiting Case" in Enumerative Geometry

Discussions: The "Limiting Case" in Enumerative Geometry

Enumeration of points, lines, planes, etc.

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $\lvert E \rvert$ lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points $E$ in a projective space determines at least $\lvert E \rvert$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of an $r$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(r-p)$-dimensional subspaces as there are $p$-dimensional subspaces, for every $p$ at most $\frac{1}{2} r$. This confirms the “top-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.