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## Über dieses Buch

Algebraic Geometry and its Applications will be of interest not only to mathematicians but also to computer scientists working on visualization and related topics. The book is based on 32 invited papers presented at a conference in honor of Shreeram Abhyankar's 60th birthday, which was held in June 1990 at Purdue University and attended by many renowned mathematicians (field medalists), computer scientists and engineers. The keynote paper is by G. Birkhoff; other contributors include such leading names in algebraic geometry as R. Hartshorne, J. Heintz, J.I. Igusa, D. Lazard, D. Mumford, and J.-P. Serre.

## Inhaltsverzeichnis

### 1. Mathematics and India

Not being qualified to lecture on a technical subject appropriate to a conference on algebraic geometry, the organizers have been kind enough to let me articulate some thoughts on the role of mathematics in Indian culture instead. In the shaping of these thoughts my contacts with promising young students in India from 1949 to 1959 have been vital. First and foremost among these students was Shreeman Abhyankar, 1949–1951. It is therefore with much pleasure that I participate on the auspicious occasion of his 60th birthday. I will deal with “Mathematics and India” from a rather special angle that brings out the difference in the roles that mathematics has played in European and Indian civilization, and focuses on research initiation in India.

P. R. Masani

### 2. Square-root Parametrization of Plane Curves

By calculating the genus of an irreducible algebraic plane curve of degree n in terms of its singularities, we see that, counted properly, the curve can have at most $$\frac{{(n - 1)(n - 2)}} {2}$$ double points, and it can be rationally parametrized iff this maximum is reached. If the maximum falls short by one or two, then the curve can still be parametrized by square-roots of rational functions. Such a square-root parametrization is used for factoring certain bivariate polynomials over a finite field.

Shreeram S. Abhyankar

### 3. A Letter as an Appendix to the Square-Root Parameterization Paper Of Abhyankar

I received two days ago your preprint “Square—root …” containing the SL(2,8)—extension in char. 7.

J.-P. Serre

### 4. Equisingularity Invariants of Plane Curves

First we want to state some questions about the classification of singularities of plane curves.

Angel Granja

### 5. Classification of Algebraic Space Curves, III

Over a fixed algebraically closed ground field k, we consider irreducible non-singular projective curves in the projective space P k 3. The original classification problem for algebraic space curves could be described as finding all such curves, giving their numerical invariants, and determining the algebraic families they belong to. This was the problem tackled in the great treatises of M. Noether [10] and G. Halphen [5] over a hundred years ago, and has been the subject of numerous investigations since then. The determination of which pairs < d, g > can occur as the degree and genus of a non-singular space curve was stated by Halphen, but only properly proved within the last decade by Gruson and Peskine [4]. Now it is reasonable to ask for finer numerical data, for example one can ask for the postulation of all possible curves, which means for each n, the number of conditions for a hypersurface of degree n to contain the curve.

Robin Hartshorne

### 6. Plane Polynomial Curves

We study some properties of generator sequences of planar semigroups and give a method of construction of plane curves with one place at infinity with given generator sequences. We also discuss similar questions for polynomial curves.

Avinash Sathaya, Jon Stenerson

### 7. A Sharp Castelnuovo Bound for the Normalization of Certain Projective Surfaces

Let P be the projective r-space (r ≥ 3) over an algebraically closed field k. Consider a reduced, irreducible, complete, non-degenerate surface X ⊆ P of degree d, and let $$\bar X$$ be its normalization: we shall assume $$\bar X$$ to be smooth (which is the case, e. g., when X is the generic projection of a given smooth projective surface $$\bar X$$). Let ∑(n) be the linear system of curves cut out on $$\bar X$$ by the hypersurfaces of P of degree n; in other words ∑(n) corresponds to the image of the canonical map $$\rho _n :H^0 (\text{P},O_\text{P} (n)) \to H^0 (\bar X,O_{\bar X} (nD))$$ (where D is the pull-back, via the normalization morphism $$\nu :\bar X \to X$$, of the generic hyperplane section X′ of X).

### 8. Abhyankar’s Work on Desingularization

It is not our aim in this article to present an exhaustive account of Abhyankar’s work related to singularities. Such an account can not be given in a short article. We shall present only a modest exposition of some of the ideas that play a central role in his resolution of singularities. Even this would be lengthy and necessarily very technical if we were to discuss the desingularization of Arithmetic Surfaces, or that of solids, or the expansions related to Canonical Desingularization. Therefore, we shall restrict ourselves mainly to the case of surfaces (over an algebraically closed ground field).

S. B. Mulay

### 9. Moduli Spaces for Special Surfaces of General Type

Classification of algebraic varieties consists of two parts: First find a set of discrete invariants like dimensions, genera, … to describe the basic topological properties of the variety, and then try to make the set of all varieties with given discrete invariants into an algebraic variety, the so called moduli space. The prototypical example is classification of Riemann surfaces: The genus is sufficient to fix the topological (and even differentiable) structure, and the set of all Riemann surfaces of a given genus can be made into an irreducible quasi-projective variety; its dimension is 3g — 3 for g ≥ 2.

Wolfgang K. Seiler

### 10. A Stationary Phase Formula for p-ADIC Integrals and its Applications

The classical stationary phase formula is for the oscillator integral: $$I(\lambda ) = \int\limits_\mathbb{R}^n {\Phi (x)\,\,\,\exp (i\lambda f(x))\,\,\,dx.}$$

Jun-ichi Igusa

### 11. The Q-admissibility of 2A 6 and 2A 7

Let L be a finite extension of the field K. Then L is K-adequate if L is a maximal subfield of a division algebra with center K.

Walter Feit

### 12. Groups Which Cannot be Realized as Fundamental Groups of the Complements to Hypersurface in C N

Finding restrictions imposed on a group by the fact that it can appear as a fundamental group of a smooth algebraic variety is an important problem particularly attributed to J.P. Serre. It has rather different aspects in characteristic p and zero and here we will address exclusively the latter case. Most restrictions described in the literature seem to rely on Hodge theory or some clever use of it (cf. [2]). A prototype of such restrictions is evenness of rk(π1/π′1 ⊗ Q)where π′1 is the commutator subgroup of the fundamental group 1rl. In the case of open non-singular varieties one can apply mixed Hodge theory. This was done by J. Morgan who obtained restrictions on the nilpotent quotients of the fundamental groups [13]. Here I shall describe a different (but also by no means complete) type of restriction on the fundamental groups of open varieties which are complements to hypersurfaces in Cn It is implicitly contained in previous work on Alexander polynomial of plane curves [4]. For example many knot groups cannot occur as fundamental groups of the complement to an algebraic curve. This gives automatically the same restrictions on the fundamental groups of complements to arbitrary hypersurfaces in Cn as follows from the well-known argument using Zariski Lefschetz type theorem: For a generic plane H relative to given hypersurface V in Cn the natural map $$\pi _1 (H - H \cap V,p_0 ) \to \pi _1 (C^n - V,p_0 )(p_0 \in H)$$is an isomorphism, i.e., possible fundamental groups of the complement to hypersurfaces in Cn are precisely the fundamental groups of the complements to plane curves. Therefore from now on I shall work with plane curves only.

Anatoly S. Libgober

### 13. Unramified Coverings of the Affine Line in Positive Characteristic

Let p be a prime number. We shall show that there is a finite unramified covering C of the affine line A1, defined over the field of p elements, so that A1 is the quotient of C by the fixed-point free action of SL n (Fpk).

### 14. Abhyankar’s Work on Young Tableaux and Some Recent Developments

This paper is meant to be a survey of Abhyankar’s work on Young tableaux and some of the subsequent developments motivated by it. Leaving the technical details to the remaining sections, we now attempt to give an overview by narrating some background, a little bit of history, and some stories in the two paragraphs below. In keeping with the spirit of Abhyankar’s work on Young tableaux, much of this paper is of an elementary nature except perhaps for a few words thrown here and there.

### 15. Abhyankar’s Recursive Formula Regarding Standard Bi-Tableau

Let X = (X ij ) be an m(l) by m(2) matrix whose entries X ij , 1 ≤ i ≤ m(1), 1 ≤ j ≤ m(2); are indeterminants over a field K. Let K[X] be the polynomial ring in these m(l)m(2) variables over K.In [1], Abhyankar enumerates standard Young bi-tableau with certain conditions and deduces that standard monomials in minors of X form a base for the vector space K[X] over K, a well known result which has been proved by DeConcini-Eisenbud Procesi using straightening formula [2]. The polynomial expression enumerating these bi-tableau involves certain integer valued functions F D (LK)(m, p, a), which characterize the Hilbert polynomial of certain determinantal ideals. Using Abhyankar’s recursive formula developed in [1], we prove certain properties of these integer valued functions.

Shirinivas G. Udpikar

### 16. Correspondences Between Tableaux and Monomials

Let X be an m(1) by m(2) matrix whose entries Xij are independent indeterminates over a field K and let K[X] be the ring of polynomials in these m(l)m(2) indeterminates. A p by p minor of X can be represented by the row indices 1 ≤ a(l, 1) < a(1, 2) < … < a(l, p) ≤ m(1) and the column n indices 1 ≤ a(2, 1) < a(2, 2) <… < a(2, p) ≤ m(2). Such a pair of strictly increasing sequences of positive integers of same (finite) length may be called a bivector whose length is p and which is bounded by m= (m(l), m(2)); such a bivector may be denoted as a(k, i)1≤k≤2,1≤i≤p ; if a′ is another bivector whose length is p′ and which is bounded by m then we write a ≤ a′ to mean that p ≥ p′ and a(k, i) ≤ a′(k, i) for k = 1, 2 and i= 1, 2,…,p′. A Young bitableau T bounded by m, i. e., a finite sequence of bivectors bounded by m, can be made to correspond to the product of the corresponding minors of X, i. e., to a certain monomial in the minors of X. This monomial is said to be standard if the bitableau T is standard, i. e., if T is a finite nondecreasing sequence of bivectors. Then we have the Standard Basis Theorem, which says that the set of standard monomials in the minors of X, corresponding to standard bitableaux bounded by m, gives a K-vector-space-basis of K[X]. The said Standard Basis Theorem was first proved by Doubilet-Rota-Stein in [13] where they call it the Straightening Law. This law has proved to be of much significance in some aspects of Algebraic Geometry and Invariant Theory.

Sanjeevani V. Vaidya

### 17. Report on the Torsion of Differential Module of an Algebraic Curve

There is a conjecture, that the torsionfreeness of the module of differentials in a point of an algebraic or algebroid curve should imply that the curve is non singular at that point. A report on the main results is given.

Robert W. Berger

### 18. A Quick Proof of the Hartshorne-Lichtenbaum Vanishing Theorem

In this note we give a short and elementary proof of the Hartshorne-Lichtenbaum vanishing theorem [8], 3.1 (which we refer to below as “HLVT”). Other authors have given nice proofs of this result, cf. [9], III, 3.1 and [4]. The reason we present another proof is threefold. First, it is an important and basic result, for example HLVT plays a crucial role in the vanishing theorems for local cohomology in [7], and in the simple proof of Falting’s connectedness theorem in [2]. Secondly we feel our proof does represent a true simplification, and thirdly the method of proof is probably useful in other contexts and is certainly suggestive of other vanishing theorems. Often a different proof is necessary for a variety of reasons, e. g. G. Lyubeznik [11] recently needed to give a new proof of HLVT (using a local Bertini theorem) to extend it to étale cohomology.

Markus Brodmann, Craig Huneke

### 19. Projective Lines Over One-Dimensional Semilocal Domains and Spectra of Birational Extensions

In [7], Nashier asked if the condition on a one-dimensional local domain R that each maximal ideal of the Laurent polynomial ring R[y, y-1] contracts to a maximal ideal in R[y] or in R[y-1] implies that R is Henselian. Motivated by this question, we consider the structure of the projective line Proj(R[s, t]) over a one-dimensional semilocal domain R (the projective line regarded as a topological space, or equivalently as a partially ordered set). In particular, we give an affirmative answer to Nashier’s question. (Nashier has also independently answered his question [9].) Nashier has also studied implications on the prime spectrum of the Henselian property in [8] as well as in the papers cited above.

William Heinzer, David Lantz, Sylvia Wiegand

### 20. Some Questions on Z[√14]

There are several definitions of Euclid rings (cf. [1]), and we recall here two of them. Throughout this article, we mean by a ring a commutative ring with identity.

Masayoshi Nagata

### 21. Function Fields of Conies, a Theorem of Amitsur—MacRae, and a Problem of Zariski

One of the first and most fundamental results in the theory of non-algebraic field extensions is Lüroth’s theorem (1876): If k ⊂ L ⊆ k(t) = K are field extensions, with t transcendental over k, then there exists u ε L such that L = k(u).

Jack Ohm

### 22. Gradings of Polynomial Rings

These brief notes have their origin in a conversation I had with Abhyankar during a pleasant walk along the Wabash a few years ago. While I tried to explain some ideas on torus actions on affine spaces Abhyankar, in that inimitable way he has to concentrate one’s mind, feigned innocence as to such fancy notions, but was quite happy to listen when I proposed talking about gradings of polynomial rings instead. The two topics cover exactly the same ground from two different points of view, equally valid and valuable. Torus actions also provide an uncomplicated introduction to the larger subject of reductive group actions. I gave my conference talk hoping that such an introduction in the readily understood language of gradings would be appreciated by some of the non-experts in the rather diverse audience Abhyankar’s birthday was bound to bring together. This is a slightly expanded version of my talk. It is entirely expository in nature and not meant for the specialists.

Peter Russell

### 23. Rigid Hilbert Polynomials for m-Primary Ideals

Which Hilbert polynomials for an m-primary ideal I in a d-dimensional, (d > 0), local Cohen-Macaulay ring (R, m) determine Heilbert function of I? For example, if we denote the Hilbert function giving the length of R/Inby H I (n) and the corresponding polynomial by p I (X), then any m-primary ideal I having Hilbert polynomial $$p_I(X) = \lambda \left( {X + \mathop d\limits_d - 1} \right)$$, has Hilbert function $$H_I = \lambda \left( {n + \mathop d\limits_d - 1} \right)$$ for all n > 0 and, in addition, I must be generated by d elements.

Judith D. Sally

### 24. One-Dimensional Local Rings with Finite Cohen-Macaulay Type

Let (R,m) be a local ring of dimension one. Recall that a non-zero finitely generated R-module M is said to be a maximal Cohen-Macaulay module (MCM module for short) provided m contains a non-zero-divisor on M. Equivalently, the simple module R/m does not embed in M. We say R has finite Cohen-Macaulay type (finite CM type) provided R has, up to isomorphism, only finitely many indecomposable MCM modules.

Roger Wiegand

### 25. Some Applications of Constructive Real Algebraic Geometry

In this short article we summarize a number of recent applications of constructive real algebraic geometry to geometric modelling and robotics, that we have been involved with under the tutelage of Abhyankar.

Chandrajit L. Bajaj

### 26. An Improved Sign Determination Algorithm

Recently there has been a lot of activity in algorithms that work over real closed fields, and that perform such calculations as quantifier elimination or computing connected components of semi-algebraic sets. A cornerstone of this work is a symbolic sign determination algorithm due to Ben-Or, Kozen and Reif [1]. In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quantifier elimination in pseudo-polynomial time. By purely symbolic, we mean that it is possible to eliminate a quantified variable from a system of polynomials no matter what the coefficient values are. The previous methods required the coefficients to be themselves polynomials in other variables. Our new method allows transcendental functions or derivatives to appear in the coefficients.

John Canny

### 27. Decomposition Algorithms in Geometry

Decomposing complex shapes into simpler components has always been a focus of attention in computational geometry. The reason is obvious: most geometric algorithms perform more efficiently and are easier to implement and debug if the objects have simple shapes. For example, mesh-generation is a standard staple of the finite-element method; partitioning polygons or polyhedra into convex pieces or simplices is a typical preprocessing step in automated design, robotics, and pattern recognition. In computer graphics, decompositions of two-dimensional scenes are used in contour filling, hit detection, clipping and windowing; polyhedra are decomposed into smaller parts to perform hidden surface removal and ray-tracing.

Bernard Chazelle, Leonidas Palios

### 28. Single Exponential Path Finding in Semi-algebraic Sets, Part II: The General Case

This paper is devoted to the following result. Let S be a semi-algebraic subset of Rn; one can decide in single exponential time whether two points of S belong to the same semi-algebraically connected component of S, and if they do, one can find a semi-algebraic path connecting them. This paper is the sequel to [HRS 4] in which the result is proved in the particular but fundamental case of a bounded regular hypersurface.

Joos Heintz, Marie-Francoise Roy, Pablo Solerno

### 29. An Improved Projection for Cylindrical Algebraic Decomposition

It is proved that the projection needed by the CAD algorithm needs only to compute the resultants, the discriminants, the leading coefficients and the constant coefficients of the input polynomials, provided they have be made square—free and relatively prime by GCD computations. This improves [7] first improvement by removing any dimension condition and dropping out from the projection set all coefficients of the input polynomials, other than the leading and the constant ones.

D. Lazard

### 30. Degree Bounds of Gröbner Bases

We show that there is no universal bound of Gröbner bases when the coefficient ring is Z. A similar result where the coefficient ring is k[z], k a field, was proved by [1]. While the coefficient ring is a field, such bounds always exist as proved by Weispfenning [16].

Wei Li

### 31. Elastica and Computer Vision

I want to discuss the problem from differential geometry of describing those plane curves C which minimize the integral $$\int\limits_C {(\alpha k^2 + \beta )ds.}$$ Here α and β are constants, kis the curvature of C, ds the arc length and, to make the fewest boundary conditions, we mean minimizing for infinitesimal variations of C on a compact set not containing the endpoints of C. Alternately, one may minimize $$\int\limits_C {k^2 ds}$$ over variations of C which preserve the total length.

David Mumford

### 32. Isolator Polynomials

This paper explores the problem of isolating the real roots of a polynomial p(x) with real coefficients, that is, of locating intervals which contain exactly one real root of p. A new solution to this problem is presented, consisting of finding a pair of auxiliary polynomials whose set of combined real roots contain at least one value in every closed interval defined by each pair of adjacent real roots in p. It is shown that any member of the polynomial remainder sequence generated by p and p′ can serve as one of these auxiliary polynomials.

Thomas W. Sederberg, Geng-Zhe Chang

### 33. A Bound on the Implicit Degree of Polygonal Bézier Surfaces

Recent work has shown that the triangular rational Bézier surface representation can be used to create polygonal surface patches. The key to the construction is the judicious use of zero weights in creating the surface parameterization. These zero weights introduce base points into the resulting rational parameterizations. Base points also lower the degree of the implicit representation of these polygonal surface patches. This paper states and proves in a simple, constructive manner a bound on the implicit degree of these rational surfaces.

Joe Warren
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