# AMS Special Session: Computational Algebraic Geometry for Curves and Surfaces

Organized by Mika Seppälä (Florida State University) and Emil Volcheck (National Security Agency).

This special session will be held Friday and Saturday, 12-13 January 2001, at the New Orleans Joint Mathematics Meetings. This is the second special session on this topic to be held at the Joint Mathematics Meetings. The first special session was held in 1999 at the San Antonio Joint Meetings.

### Scope

This session is devoted to algorithms and constructive techniques for algebraic curves, Riemann surfaces, and algebraic surfaces. We are interested in reports on algorithms to solve problems or on a significant use of computational algebra techniques to obtain results. Algorithmic, algebraic, arithmetic, and analytic aspects of curves and surfaces are appropriate topics.

Read about the 1999 special session on computational algebraic geometry for curves and surfaces to see what previous speakers in this series have presented.

The deadline to guarantee consideration of an abstract was 8 August 2000.

The final deadline for abstract submission is 3 October 2000. We are accepting new proposals on a rolling basis up to the final deadline, as space permits.

### Submission

Please write to both of us, Mika Seppälä at seppala@math.fsu.edu and Emil Volcheck at volcheck@acm.org, to propose a topic or to inquire whether a topic would be appropriate. We encourage you to send us an abstract of your talk and/or a URL of a paper or manuscript on which you would like to speak.

To formally submit an abstract to the AMS, read the guidelines and submit your abstract.

## Speakers

This list is current as of 8 January 2001. There are twenty-two (22) speakers in total.

## Abstracts

### Ruled surfaces contained in quadric cones in five dimensional complex projective space

Gian Mario Besana, Alberto Alzata, Marina Bertolini

A projective variety $X$ is called {\it projectively normal} if its homogeneous coordinate ring is integrally closed. More geometrically, this means that hypersurfaces of any degree $t \ge 1,$ in the ambient projecive space, cut on $X$ complete linear systems. The hardest step in establishing the projective normality of a variety is usually the case $t=2.$ This question can be addressed by understanding how many quadric hypersurfaces contain the given variety. In this context, the present paper examines ruled surfaces in $\mathbb{P}^5$ which are contained in singular quadric hypersurfaces. A complete classification is given for ruled surfaces contained in quadrics of rank $5$ and a series of general results are found in the cases of lower ranks. Part of the classification is achieved by first finding upper bounds for a list of numerical invariants and subsequently having Maple perform the remaining finite number of numerical checks.

### Computing Conformal Mappings of Curved Surfaces

We survey some of the practical computational issues that arise from our work with an interdisciplinary team of mathematicians and neuroscientists involved in an effort to build a reliable tool for approximating conformal flat mappings of the human brain. Of course, our method actually works to approximate conformal mappings of arbitrary curved surfaces as well as conformally correct shapes of conformal tilings. Circle packings are used to provide the approximations. This is joint work with a team that includes Ken Stephenson, DeWitt Sumners, and Monica Hurdal.

### Short Homology Bases of Compact Riemann surfaces

This is ongoing work with Mika Seppälä (FSU). The goal is to find canonical homology bases represented by simple closed geodesics which are as short as possible. Pants decompositions as well as geodesic triangulations are used to reduce the geometric problem to a combinatorial one. In computational Riemann surface theory, this can be used to obtain short input parameters.

### The Use of Computer Algebra Programs for the Classification and for Explicit Constructions in the Theory of Algebraic Surfaces

I will report on joint work with Roberto Pignatelli, Ingrid Bauer and Fabio Tonoli, centered on explicit calculations of

1. moduli spaces of Godeaux surfaces
2. canonical rings of surfaces whose canonical system has base points
3. relative canonical rings of fibrations of small genus
4. existence problems for even sets of nodes.
The problem faced is rather general: what is the best theoretical set up in which moduli spaces can be effectively described, or existence questions resolved?

### Resultants and Chow forms via Exterior Algebra

I will describe recent work with Frank-Olaf Schreyer in which we show how to compute Chow forms (and in particular, resultants) via free resolutions over the exterior algebra. The method works particularly well for curves, where it gives matrices in the Pluecker coordinates whose determinants are expressions of the Chow form. The matrices may be computed in terms of free resolutions over the exterior or over the symmetric algebra.

### Specializing $A_n$ Covers of Elliptic Curves with $j$-invariant 1728

This talk will deal with realizing groups as Galois groups over the rational numbers by specializing Galois covers of elliptic curves. In previous work, we used a variant of Hilbert's irreducibility theorem for elliptic curves of positive rank to realize the alternating group $A_n$ as a Galois group over $\mathbb{Q}$ when $n\ge{5}$ is not divisible by 3. In particular, we showed that each of these groups could be realized by specializing Galois covers of elliptic curves with $j$-invariant 0. In this talk, we will look at some recent progress in trying to realize $A_n$ as a Galois group over $\mathbb{Q}$ by specializing Galois covers of elliptic curves with $j$-invariant 1728. Assuming the Birch Swinnerton-Dyer conjecure and using the root number formula of Birch-Stephens, we are able to give some sufficient conditions on $n$ for $A_n$ to be realizable as a Galois group by specializing such a cover.

### An Algorithm for Symplectic Matrices of Finite Order Surface Mapping-classes

Moira Chas, Jane Gilman

Assume that $h$ is a conformal automorphism of a compact Riemann surface $S$ of genus $g \ge 2$. Equivalently $h$ can be thought of as a representative of a finite order mapping-class. If $h$ is of prime order $p$ with $t$ fixed points, then $h$ is determined up to conjugacy by a $(p-1)$-tuple of integers $(n_1, ...., n_{p-1})$ where $t = \Sigma_{i=1}^{p-1} n_i$ and $\Sigma_{i=1}^{p-1} i*n_i \equiv 0 (p)$. We present an algorithm whose input is the $(p-1)$-tuple of integers and whose output is the symplectic matrix of the action of $h$ on a maximally adapted canonical homology basis. Previous results gave a non-symplectic matrix representation for $h$ on an adapted basis together with the intersection matrix. We discuss settings in which the algorithm can be implemented.

### Riemann Surface Calculation of Nonlinear Electrical Filters

Martin Hassner, Barry Trager

We solve an Inverse Spectral Problem defined on an elliptic Riemann surface. For a given sum of two elliptic integrals of the third kind, which is used as a spectral measure, we calculate a nonlinear finite difference operator which can be realized by electronic components. This spectral measure appears as the magnetostatic potential of a magnetic signal read sensor.

The practical application of such filters, matched to a potential function defined on the elliptic Riemann surface, is the decomposition of arbitrary sums of shifted magnetostatic potentials into elementary components. Furthermore, if these sums are constrained, these filters generate phase shifts that allow to check the imposed constraint.

The voltages and currents measured in the electronic circuit which realizes the nonlinear finite difference equation are the scattering data for an inverse spectral transform that reconstructs the input potential function.

### Milagros Izquierdo

Antonio Costa, Milagros Izquierdo

### On the locus of real Riemann surfaces

It is well known the functional equivalence between pairs $(X,\sigma)$,
where $X$ is a Riemann surface which admits an antiholomorphic
involution (symmetry) $\sigma : X \to X$, and real algebraic curves. We
shall refer to such Riemann surfaces as real Riemann surfaces. We show
by means of the universal covering tranformation groups and their
Schereier graphs that any real Riemann surface can be quasiconformally
deformed to a real Riemann surface $\sigma : X \to X$ such that $X$
admits a symmetry $\tau$ which fixes one non-separating curve. As a
consequence we give a distinct proof of the connectedness of the subset
of real Riemann surfaces in the moduli space of Riemann surfaces of
given genus of the one given by Buser, Sepp{\"a}l{\"a} and Silhol.

### Computing Teichmuller Distance Between Elliptic Curves

Eric P Klassen, Craig Nolder, Mika Seppala, Tyler Sutton

We give computer implementations of the following operations involving elliptic
curves: (1) Given an algebraic equation for an elliptic curve $X$, compute a
reduced lattice $\Lambda$ such that $X=C/\Lambda$. (2) The inverse of (1):
Given a lattice for $X$, compute an algebraic equation. (3) Given two elliptic
curves (presented either by equations or lattices), compute the Teichmuller
distance between them.

### Reduction of Wildly Ramified Covers of Curves

Let R be a finite extension of W(k) the ring of Witt vectors over a field k of positive characteristic p. Let K=frac(R). We consider p-cyclic (wildly) ramified covers X of the projective line over K and ask when does X have good reduction or more generally what is its stable reduction. In some cases the answers can be given explicitly in terms of the branch cycle description of the cover; in general the results will be algorithms. The strategy is to start with an R-model for the projective K-line and to find its normalization in the function field of X, i.e., finding equations for it.

### Macaulay Resultant of Composed Polynomials

The main question of this paper is: What is the Macaulay resultant of composed polynomials? By a composed polynomial $f\circ(g_1,\ldots,g_n)$, we mean the polynomial obtained from a polynomial $f$ in the variables $y_1,\ldots,y_n$ by replacing $y_j$ by by some polynomial $g_j$. Cheng, McKay and Wang and Jouanolou have provided answers for two particular subcases. The main contribution of this paper is to complete these works by providing a uniform answer for all subcases. In short, it states that the Macaulay resultant is the product of certain powers of the Macaulay resultants of the component polynomials and of some of their leading forms. It is expected that these results can be applied to compute Macaulay resultants of composed polynomials with improved efficiency. We also state a lemma of independent interest about the Macaulay resultant under vanishing of leading forms.

### Hyperspheres and Hyperplanes Fitted Seamlessly by Algebraic Constrained Total Least-Squares

For each finite set of points in a Euclidean space of any dimension, the algorithm presented here determines all the algebraically best fitting circles or lines, spheres or planes, or hyperspheres or hyperplanes, in a seamless manner from spherical through affine manifolds. In particular, affine submanifolds of any dimensions are {\it not\/} singularities of the algorithm. To this end, the algorithm combines projective geometry, Coope's and Gander, Golub, and Strebel's layouts of the equations, and Golub, Hoffman, and Stewart's generalization of the Schmidt-Mirsky matrix approximation theorem to solve the equations. The resulting best fitting manifolds remain invariant under rigid transformations. Moreover, if the best fitting manifold is affine, then it coincides with Golub and Van Loan's affine manifold of Total Least-Squares. Thus the algorithm can also fit hyperspheres in a manner that remains robust with data lying near a hyperplane.

### Computations with Tori II

Craig Nolder, Eric P Klassen, Mika Seppala, Tyler Sutton

We continue our investigation of computationally identifying algebraic curves and Riemann surfaces. We address the problem of calculating the Teichmuller distance between two algebraic curves.

### Hyperbolic Lego and Algebraic Curves

The inverse of the uniformization problem is to find an equation for the algebraic curve associated to a given hyperbolic surface. We will show how by attaching certain conformal invariants to hyperbolic polygons one can solve this problem for some families of surfaces obtained by assembling such polygons.

### Maximally Inflected Curves

Viatcheslav Kharlamov, Frank Sottile

A map from P^1 to P^n of degree d necessarily has (n+1)(d-n+1) points where it is ramified, counted with multiplicities. We call these inflection points, as in the image curve, a point of simple ramification is an inflection point. A maximally inflected curve is a real curve, all of whose inflection points are also real. The existence of such curves is guaranteed by a result in the real Schubert calculus, and these curves are closely related to an important conjecture in that field. Maximally inflected plane curves satisfy some topological restrictions given by the Klein and Pluecker formulas. They also satisfy some more subtle restrictions whose existence was discovered experimentally. This talk will introduce these objects, then discuss symbolic and numerical techniques to generate examples of maximally inflected curves, and lastly describe the known topological restrictions, including those whose existence is only suspected via experiments.

### Computation of Monodromies and Bers Embedded Teichmueller Spaces for Once Punctured Tori

Yohei Komori, Toshiyuki Sugawa, Masaaki Wada, Yasushi Yamashita

Let $X$ be the once punctured torus given by the quotient $(\mathbb{C}\setminus L)/L,$ where $L$ is the additive lattice group generated by $1$ and $\tau$ over the integers for a point $\tau$ in the upper half plane. By the classical correspondence, we see that the four-times punctured sphere $Y=\mathbb{C}\setminus \{0,1,\lambda\}$ is commensurable with $X,$ where $\lambda$ is the value of the elliptic modular function at $\tau.$ For a bounded projective structure (holomorphic quadratic differential) on $X,$ we can compute the monodromy by numerically solving the differential equation
$$2y''+\left( \frac{t+c(\lambda)}{z(z-1)(z-\lambda)}+\frac{1}{2z^2(z-1)^2} +\frac{1}{2(z-\lambda)^2}\right) y=0$$
on $Y,$ where $t$ is a parameter corresponding to the projective structure and $c(\lambda)$ is the accessory parameter determined by $\lambda.$

By using this method, we compute pleating rays corresponding to simple closed geodesics in $X,$ and then visualize the shape of the Bers embedded Teichm\"uller space of $X$ in the $t$-plane. As a by-product, we can compute the accessory parameter $c(\lambda)$ numerically. The above method also enables us to visualize the discreteness locus of monodromy groups by employing the method developed by Yamashita and Wada.

### Visualization of Riemann Surfaces

Since more than 100 years Riemann surfaces are an important concept in mathematics and enjoy today an increasing popularity in modern theoretical physics (string theories, conformal field theories). Faithful representations of Riemann surfaces of simple functions can be found in form of plaster and wood models in many mathematics department. Using symbolic manipulation programs, it is today possible to generate visualizations of virtually every multi-valued function. The outline of implementations of programs for the automatic generation of Riemann surfaces of arbitrary algebraic functions and arbitrary compositions of elementary functions and selected special functions of mathematical physics are given. Examples of pictures of Riemann surfaces of many classes of functions are shown. Pictures of Riemann surfaces of most elementary and special functions will be available soon at http://www.functions.wolfram.com.

### Computing the Period Matrix of an Algebraic Curve

A black-box program for the explicit calculation of the Riemann matrix (the period matrix) of arbitrary compact connected Riemann surfaces is presented. All such Riemann surfaces are represented by the equation for a (possibly singular) plane algebraic curve. The method of calculation of the Riemann matrix is essentially its definition: we numerically integrate the holomorphic differentials of the Riemann surface over the cycles of a canonical basis of the homology of the Riemann surface. Both the holomorphic differentials and the canonical basis of the homology of the Riemann surface are obtained exactly through symbolic calculations. This program is included in Maple 6, as part of the algcurves package.

### Numerical Uniformization of Hyperelliptic Curves

We consider a closed, hyperelliptic curve $C$ defined by $w^2=p(z)$. The goal is to represent $C$ as the quotient of the hyperbolic plane by the action of a fuchsian group. First we determine the hyperbolic metric on $C$ numerically by integrating the curvature equation $\Delta u = e^{2u}$. Then we use a relation between the hyperbolic metric and the uniformizing projective connection on $C$ to calculate the so called accessory parameters. Once these are known, we obtain a uniformizing representation $\varrho: \pi_1(C)\longrightarrow PSL_2(\Bbb R)$ by integrating a differential equation of the form $y''+\frac{S}{2}y$ along representative elements of $\pi_1(C)$.

### Deformation of Packable Surfaces

Roger Barnard, Brock Williams

A circle packing is a configuration of circles with a prescribed pattern of tangencies. This prescribed pattern'' is a purely combinatorial object usually encoded in a graph or abstract triangulation. Beardon and Stephenson showed that for any reasonable triangulation of a surface $S$, there is a unique conformal structure on $S$ that supports a packing with the given pattern of tangencies. We explore how the conformal structure varies with the triangulation. In particular, we discuss the effect of combinatorial earthquakes and welding deformations. Using these operations, any packable Riemann surface of type $(g,n)$ can be deformed so as to approximate any other given surface.

### An algorithm deciding the discreteness of a given Mobius group generated by two loxodromic elements with parabolic commutator

Yohei Komori, Toshiyuki Sugawa, Masaaki Wada, Yasushi Yamashita

Let $F_2 = < a , b >$ be a free group with two generators $a, b$, and $\rho : F_2 \to \mbox{PSL}(2, C)$ a strictly type preserving representation, i.e.
$\mbox{tr}[\rho(a), \rho(b)] = -2$. We denote by $X$ the set of all such $\rho$
up to conjugacy. If the image $\rho(F_2)\subset \mbox{PSL}(2, C)$ is discrete,
it is called a {\em punctured torus group\/} and studied by many people. But,
in general, it is very difficult to decide the discreteness of a given subgroup
of $\mbox{PSL}(2, C)$. Our result is that there is an effective method so
that we can show the discreteness or indiscreteness of the group $\rho(F_2)$
for almost all $\rho\in X$. We have a computer program of this algorithm and
produced some pictures of discrete loci for several slices of $X$. One of our
products is the pictures of Bers embedding of the Teichm\"uller spaces for once
punctured tori. We can expect to observe a self-similarity of the boundary of a
Bers slice by our computation. There are some previous works for linear slices
of $X$. The reader can compare the pictures of McMullen, Wright and ours at
McMullen's Web site
(http://abel.math.harvard.edu/~ctm/gallery.html)

## Schedule

Friday 8-11 AM (B1) (analytic)

08:00 Izquierdo 962-14-1070
08:30 Buser 962-30-747
09:00 Gilman 962-20-988
09:30 Silhol 962-14-919
10:00 Wagner 962-30-930
10:30 Williams 962-30-103

Friday 1-6 PM (B2) (applications, visualization, software packages)

1:00 Trott 962-97-1067
1:30 Klassen 962-51-1275
2:00 Sugawa 962-30-748
2:30 Minimair 962-14-659
3:00 Van Hoeij 962-14-1009
3:30 Hassner 962-14-1399
4:00 Nievergelt 962-65-120
4:30 Nolder 962-14-1340
5:00 Yamashita 962-30-1012
5:30 Bowers 962-30-979

8:30 Special Session Dinner at Mat & Naddie's Café

Saturday 8-11 AM (B3) (algebraic and arithmetic)

08:00 Besana 962-14-134
08:30 Evans 962-11-657
09:00 Catanese 962-14-724
09:30 Eisenbud 962-14-779
10:00 Sottile 962-14-96
10:30 Lehr 962-14-815

## Dinner

Participants in the special session and friends are invited to join us for dinner on Friday, January 12, at 8:30 PM at Mat & Naddie's Café (see http://neworleans.citysearch.com/profile?fid=2&id=4430745.)
Address: 937 Leonidas St, New Orleans 70118, tel: (504) 861-9600