By Tor Dokken (auth.), Bert Jüttler, Ragni Piene (eds.)

The ?elds of Geometric Modeling and Algebraic Geometry, even though heavily - lated, are routinely represented by means of virtually disjoint scienti?c groups. either ?elds take care of items de?ned through algebraic equations, however the gadgets are studied in numerous methods. whereas algebraic geometry has constructed extraordinary - sults for realizing the theoretical nature of those items, geometric modeling makes a speciality of functional purposes of digital shapes de?ned by means of algebraic equations. lately, despite the fact that, interplay among the 2 ?elds has inspired new study. for example, algorithms for fixing intersection difficulties have bene?ted from c- tributions from the algebraic part. The workshop sequence on Algebraic Geometry and Geometric Modeling (Vilnius 1 2 2002 , great 2004 ) and on Computational tools for Algebraic Spline Surfaces three (Kefermarkt 2003 , Oslo 2005) have supplied a discussion board for the interplay among the 2 ?elds. the current quantity offers revised papers that have grown out of the 2005 Oslo workshop, which used to be aligned with the ?nal overview of the ecu venture GAIA II, entitled Intersection algorithms for geometry dependent IT-applications four utilizing approximate algebraic tools (IST 2001-35512) .

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Aries et al. 2. List of the covariants presented in the paper. ∂0 f0 1 ∂0 f1 Φ1 = 8 ∂0 f2 ∂0 f3 ∂1 f0 ∂1 f1 ∂1 f2 ∂1 f3 ∂2 f0 ∂2 f1 ∂2 f2 ∂2 f3 y0 y1 . 12) ∂ Here ∂i stands for dx . i This covariant Φ1 has degree 3 and type Pol3 (C3 , (C4 )∗ ). The geometric object associated to Φ1 (f ) is a parameterization of the dual surface to S(f ). Plane spanned by the image of a line. Consider a generic line L in CP2 , given by an equation λ(x) = λ0 x0 + λ1 x1 + λ2 x2 = 0. 13) Its image under f is a conic in CP3 , spanning a plane, that is an element of (CP3 )∗ .

Ruberman. A Sextic Surface cannot have 66 Nodes. J. , 6(1):151–168, 1997. 13. V. Kharlamov. Overview of topological properties of real algebraic surfaces. AG/0502127, 2005. 14. O. Labs. Algebraic Surface Homepage. Information, Images and Tools on Algebraic Surfaces. net, 2003. 15. O. Labs. A Septic with 99 Real Nodes. AG/0409348, to appear in: Rend. Sem. Mat. Univ. , 2004. 16. O. Labs. Dessins D’Enfants and Hypersurfaces in P3 with many Aj -Singularities. AG/0505022, 2005. 17. Y. Miyaoka. The Maximal Number of Quotient Singularities on Surfaces with Given Numerical Invariants.

As a consequence, it deﬁnes a function on U. 3 shows its values. 3. Discrimination between the orbits. It is already an interesting result that the inertia of one quadratic form attached to f is enough to discriminate between the six orbits in U. Now, we want to go further and deﬁne the orbits by equations and inequalities. For this we introduce the characteristic polynomial of M (f ): det(t · I − M (f )) = t3 + A1 (f ) t2 + A2 (f )t + A3 (f ). 34) Any condition on the inertia can be translated into equations and inequalities involving the coefﬁcients of Ai (f ).