Algebraic surface: F(x,y,z) = 0 p a = P 2 = 0

1. Overview Algebraic surface: F(x,y,z) = 0 pa = P2 = 0 1 2 Enriques, 1895 (over any field, inconstr.) Schicho, 1996 (over any field, constr.) Castelnuovo, 1896 (over C, inconstr.) pencil of rational curves Riemann, 1850 Noether, 1870 Sendra,Winkler, 1997 Del Pezzo surface Cayley, 1869 Del Pezzo, 1887 Schläfli, 1863 (over C) Comesatti, 1907 Manin, 1965 Bajaj/Holt/ Netravali 1997 (over R) 2a Tubular surface: A(z)x2+B(z)y2+C(z) = 0 Noether, 1870 Tsen (over C) Peternell, 1997 Schicho, 1998 (over R) 3 inconstructive (existence) (s,t)-plane constructive (algorithm)

2. The Problem Given a polynomial equation F(x,y,z)=0 in three variables we look for the parametrization of the solution set in terms of rational functions in two variables: (x,y,z) = (X(s,t), Y(s,t), Z(s,t)). In case of a linear equation like x+y+z=1 the parametrization is trivial, in this case: (x,y,z) = (s,t,1-s-t). For the sphere with equation x2+y2+z2=1 we get via the stereographic projection the parametrization: 2s 2t s2+t2-1 (x,y,z) = ( , , ) s2+t2+1 s2+t2+1 s2+t2+1 The problem of rational surface parametrization has been investi-gated for about 150 years. It soon turned out that not all surfaces can be parametrized by rational functions and that also the underlying field plays an important role. In 1896 Castelnuovo gave a necessary and sufficient condition for parametrizability over the complex that can be easily extended to a condition over the reals. For many special surfaces strategies for finding a rational parametrization have been developped, but a general solution was still missing. One of the authors filled the important gap and presented a complete algorithm for parametrizing algebraic surfaces for the first time (J.Schicho, “Rational Parametrization of Algebraic Surfaces”, 1996). In this poster we want to give an overview of the algorithm, its main tool, namely adjoints, and recent advances in terms of efficiency.

3. The Algorithm Input: polynomial F(x,y,z) Output: either “not parametrizable”, or rational functions X(s,t), Y(s,t), Z(s,t) s.t. F(X(s,t),Y(s,t),Z(s,t)) = 0 1 Compute pa and P2 . (the arithmetical genus and the second plurigenus) If (pa0 or P2>0) return “not parametrizable” . (Castelnuovo’s criterion) 2 Transform F(x,y,z) by a birational substitution (x’,y’,z’)=(sx(x,y,z),sy(x,y,z),sz(x,y,z)) to F’(x’,y’,z’), which is either a tubular or a Del Pezzo surface. (Schicho’s constructive version of Enrique’s theorem) 2a If F’(x’,y’,z’) is a Del Pezzo surface, then transform it to a tubular surface. (theory of Cayley, Del Pezzo, Schläfli, Comesatti and Manin) 3 Parametrize the tubular surface, giving (X’(s’,t’),(Y’(s’,t’),Z’(s’,t’)). (over C: theory of Noether, Tsen; over R: algorithm of Peternell/Schicho) 4 Reparametrize to get the parametrization (X(s,t),(Y(s,t),Z(s,t)) of the original surface.

4. Castelnuovo’s Criterion 1 Birational invariants are properties that do not change under birational transformations. Surfaces have infinitely many birational invariants, among them the arithmetical genus pa and the plurigeni Pm, m  0. Over the reals the number of connected components (in the projective setting) is another birational equivalence. Since parametrizibility is a synonym for birational equivalence to the plane, we get arbitrary many necessary conditions for parametrizibility: Any birational invariant of a parametrizible surface must take the same value as for the plane. In 1896 Castelnuovo could show that the coincidence of only two birational invariants (namely the arithmetical genus pa and the second plurigeni P2) is already sufficient for parametrizibility over the complex: Theorem (Castelnuovo’s criterion): A surface is parametrizible over the complex iff pa = P2 = 0. Over the reals also the number of connected components must be considered: Theorem (Real Parametrizibility): A surface is parametrizible over the reals iff pa = P2 = 0 and there is only one component (in the projective setting). The computation of pa and P2 is done via adjoints: Pm = dim(V0,m) pa = d + 2 dim(V1,1) - dim(V2,1) - 1 where d is the degree of the surface and Vn,mis the vector space of m-adjoints of degree at most n+m(d-4).

5. Schicho’s Transformation 2 New! • In 1895 Enriques could show that any parametrizable surface either has a pencil of rational curves or can be birationally transformed to a Del Pezzo surface. In 1996 Schicho could give an algorithm for this decision and transformation in his PhD-thesis for the first time. The computation again involves adjoints. • The full statement of Schicho’s theorem and algorithm would be too involved, so we just characterize the two cases. • Pencil of Rational Curves: “Rational” is a synonym for “parametrizable”, so we are talking about parametrizable curves lying on surfaces. A surface is said to have a pencil of rational curves if it is the union of rational curves that depend on one free parameter. A uniform parametrization of the curves in the pencil yields immediately a parametrization of the surface. • Del Pezzo Surfaces: Typical examples of Del Pezzo surfaces are nonsingular cubic surfaces and the nonsingular intersection of two quadrics in P4. The precise definition is the following: “A surface is called a Del Pezzo surface iff its generic plane section is elliptic and it has irregularity zero.” Here are some properties of Del Pezzo surfaces: • Del Pezzo surfaces have a degree of maximal 9. • Del Pezzo surfaces have at most 5 connected components. • Connected Del Pezzo surfaces can be parametrized. • Over the complex Del Pezzo surfaces always have a pencil of rational curves.

6. Adjoints - The Main Tool • The construction of a pencil of rational curves, the construction of birational reduction to a Del Pezzo surface and the computation of the birational invariants for Castelnuovo’s criterion (the numbers pa and P2) can be accomplished with one single tool, namely adjoints. • The concept of adjoints was developped in the 19th century, but was more or less forgotten in this century. Adjoints of a surface are other surfaces that pass through the singularities of the given surface with a certain order. More precisely: • Definition (Adjoints): A surface is an m-adjoint of a given surface iff its defining polynomial • vanishes with order at least m(r-1) at each r-fold singular curve and • vanishes with order at least m(r-2) at each r-fold singular point. • Infinitely near singularities have to be taken into account. • The adjoints of a surface clearly form an ideal. When restricted to a certain degree the adjoints form a vector space. • Adjoints are closely linked to differential forms. A rational differential form is said to be of the first kind iff its integral over any compact set is finite. With that notion we have the following: • Theorem (“Secret Definition of Adjoints”): Let F(x,y,z) be the defining equation of the surface S. Then G(x,y,z) is a 1-adjoint of S iff • Gdxdy • F/z • is a differential form of the first kind.

7. Computing Adjoints The computation of adjoints is the most expensive part of the parametrization algorithm. The problem is that also “infinitely near” singularities must be considered, i.e. singularities that coincide must be handled separately. Traditional approach: Since adjoints of a non-singular surface are trivial to compute (any polynomial is adjoint to a non-singular surface), it is a natural idea to resolve the singularities by “blowing up” and compute the adjoints via the desingularization. This approach is based on Hironaka’s theorem of existence (1966), Villemayor’s constructive version (1989) and Schicho/Bodnar’s implementation (1997). However, experiments showed that this approach is too expensive to be practial. New approach: Certain (“quasi-ordinary”) singularities can also be examined by Puiseux-series expansion. In order to transform a surface to the quasi-ordinary case only the curve singularities of the discriminant need to be resolved, which is a far simpler task. An implementation in Maple is under development (see a forth-coming paper by Schicho/Stöcher). Interactive experiments show that the performance will be much better.

8. Terminology Adjoints: Adjoints of a surface are other surfaces that have a contact of at least a certain order at the singularties of the given surface; the demanded minimal order of contact depends on the kind and the order of each singularity. Birational: A variable transformation is said to be birational iff it and its inverse both can be expressed by rational functions. Parametrizable: Whenever we say that a surface is parametrizable, we actually mean parametrizable in terms of rational functions. Proper Parametrization: A proper parametrization is an injective parametrization. For instance, the stereographic projection induces a proper parametrization of the sphere. The common (although not rational) parametrization by trigonometric functions is improper. A synonym for proper parametrizable is “rational”. Surface: The term “surface” is always used in the sense of an algebraic surface, i.e. the zero-set of a polynomial in three variables. Whenever the underlying field plays an important role, this is mentioned.

9. Example Consider the surface given by F =4x4 +8x3y+x2y2 +8x2 -7xy-y2 +xyz2-x2z2 Since pa=P2=0 we know that F is parametrizable. Schicho’s transformation algorithm detects a pencil of rational curves on F and yields a simple transformation to a tubular surface: y  z’x, z  y’ F’ = (z’2 +8z’+4) x2 + (z’-1) y’2 - z’2 -7z’+8 Computing a parametrization of F’ and transforming it back to the original variables yields: (x,y,z) = ( , , ) When plugging this parametrization in, F indeed vanishes. 1 2 3 4 t2+7t-s2t+s2-8 t3+7t2-s2t2+s2t-8t t2-10st+20t+s2t+12s2-68s+96 t2-10st+20t+s2t+12s2-68s+96 -5t2+2st2-74t-5s2t+40st+192s-34s2-272 t2-10st+20t+s2t+12s2-68s+96

10. Josef.Schicho@risc.uni-linz.ac.at www.risc.uni-linz.ac.at/projects/basic/adjoints/[param/] Automated Parametrization of Algebraic Surfaces RISC Linz J. Kepler University Linz, Austria/Europe Wolfgang.Stoecher@risc.uni-linz.ac.at