Algebraic surface: F(x,y,z) = 0

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Overview
Algebraic surface F(x,y,z) 0 pa P2 0
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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)x2B(z)y2C(z) 0
Noether, 1870 Tsen (over C)
Peternell, 1997 Schicho, 1998 (over R)
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inconstructive (existence)
(s,t)-plane
constructive (algorithm)
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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
xyz1 the parametrization is trivial, in this
case (x,y,z) (s,t,1-s-t). For the sphere with
equation x2y2z21 we get via the stereographic
projection the parametrization
2s
2t
s2t2-1
(x,y,z) ( , ,
)
s2t21
s2t21
s2t21
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.
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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 P2gt0)
return not parametrizable . (Castelnuovos
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. (Schichos
constructive version of Enriques 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.
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Castelnuovos Criterion
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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 (Castelnuovos 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,m is the vector space of
m-adjoints of degree at most nm(d-4).
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Schichos Transformation
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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 Schichos 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.

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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 Castelnuovos 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.

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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 Hironakas theorem of existence
(1966), Villemayors constructive version (1989)
and Schicho/Bodnars 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.
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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.
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Example
Consider the surface given by F 4x4 8x3y
x2y2 8x2 -7xy -y2 xyz2-x2z2 Since paP20 we
know that F is parametrizable. Schichos
transformation algorithm detects a pencil of
rational curves on F and yields a
simple transformation to a tubular surface y ?
zx, z ? y F (z2 8z4) x2 (z-1) y2 -
z2 -7z8 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.
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t27t-s2ts2-8
t37t2-s2t2s2t-8t
t2-10st20ts2t12s2-68s96
t2-10st20ts2t12s2-68s96
-5t22st2-74t-5s2t40st192s-34s2-272
t2-10st20ts2t12s2-68s96
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www.risc.uni-linz.ac.at/projects/basic/adjoints/p
aram/
Josef.Schicho_at_risc.uni-linz.ac.at
Automated Parametrization of Algebraic Surfaces
RISC Linz J. Kepler University Linz,
Austria/Europe
Wolfgang.Stoecher_at_risc.uni-linz.ac.at