On the classification of real algebraic curves and surfaces

Transcription

1 On the classification of real algebraic curves and surfaces Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo COMPASS, Kefermarkt, October 2,

2 Background In the GAIA and GAIA II projects we asked to what extent a classification of possible self intersections and other singularities of curves and surfaces could be useful for CAGD purposes. The partners responsible for this investigation are UO, UNSA, and Cantabria. Initiated by Cathrine Tegnander, then SINTEF, now Trondheim Master degree students who have participated: Rolv Breivik, Margrethe Naalsund, Wenche Oxholm, Heidi Sundby, Magnus Løberg. 2

3 History In the old days, algebraic geometry and real algebraic geometry were the same thing. Examples of 19th century work are the studies of real quartic curves by Zeuthen and of the lines on the cubic surface by Schläfli. One of Hilbert s famous problems, the 16th, was the... investigation as to the number, form, and position of the sheets which a surface of the 4th order in three dimensional space can really have. 3

4 For higher degree curves and surfaces the problems became more (too) difficult, and, as we know, these problems were solved by allowing complex solutions (and equations) as well as working in projective, rather than affine, spaces. In the 20th century some people continued to work in real algebraic geometry Hilbert s problem was solved for degree 4 in 1972 by Kharlamov but a lot of that work was local, e.g. studying singularities, developing a theory of semialgebraic varieties or real analytic or Nash manifolds... When CAGD entered the scene, the topic became hotter but there seems to be still a long way to go. 4

5 How can complex projective algebraic geometry be useful when what we want to study are real affine algebraic varieties? Of course the methods are essentially the same... the real affine variety is contained in the complex projective variety the singularities of the real variety are bounded by the singularities of the complex variety 5

6 But there are many classification problems which cannot be determined simply this way, e.g. the number of connected components of the real variety questions of boundedness equidimensionality which real types of singularities occur for each complex type 6

7 Let us look at a complex parameterized curve f : P 1 C P2 C given by three homogeneous polynomials of degree d, f = ( f 0 (s, t); f 1 (s, t); f 2 (s, t) ). The curve can be viewed as a projection of the rational normal curve of degree d in P d C. To determine the number and types of singularities and inflection points of the plane curve, it suffices to know the position of the projection centre (a linear space of dimension d 3) with respect to the family of osculating flags and the secant varieties of the rational normal curve. If the projection centre is as general as possible, the projected curve will have (d 1)(d 2)/2 ordinary double points (nodes) and 3(d 2) (ordinary) flexes. 7

8 Consider a curve in P 3 C and project it to P2 C. In order to see what kind of singularities we get, we need to know the position of the centre of projection a point with respect to, say, the tangent developable. The existence of curves with given types of singularities can be shown this way for example, if you want a plane curve with no self intersections, all the singularities must be cusps, and this means the tangent developable must have a singular point of high multiplicity. This is an additional reason for studying the singularities of a tangent developable (cf. Pål s talk). 8

9 If C is a curve in P n C k n 1 is equal to of genus g, then the total stationary index (n + 1)d + ( n + 1) (2g 2) = (n + 1)(d + n(g 1)). 2 This index is equal to the number of zeros of the (global) Wronski determinant of the family of local parameterizations of the curve, which again is equal to the degree of the nth order jet bundle of the curve (or the vector bundle PC n (1) of principal parts of order n of f O(1)). Therefore k n 1 is the degree of the line bundle Λ n+1 PC n(1). 9

10 Maximally inflected real rational curves We have seen that a parameterized curve (g = 0) of degree d in n-space has a total number of (n + 1)(d n) inflection points (this includes cusps, flexes, points of hyperosculation points that are not of type (1,2,3)). The curve is called maximally inflected if all these points are real, i.e., if all zeros of the Wronski determinant are real (cf. Kharlamov and Sottile, Shapiro and Shapiro conjecture,...). For a general (nonsingular) complex plane curve of degree d, the number of inflection points is equal to 3d + 2g 2 = 3d(d 2) (these are the intersection points of the curve and its Hessian). Zeuthen and Klein showed that at most a third of these are real! 10

11 Classification of real plane curves All real cubic plane curves and all real rational quartic curves have been classified (Zeuthen,..., Wall,..). For higher degrees, the number of cases starts exploding. From a CAGD point of view, it seems more reasonable to decide what kind of higher degree curves one needs before trying to do more of this kind of classification. For example, one could concentrate on parameterized curves. On the other hand, from a local point of view, plane curve singularities are well understood and can be completely described in the complex case via certain numerical invariants (computable from the local equation of the curve). In the real case one needs a refinement of each complex type, e.g. a singularity of type A 2k 1 can have none or two real branches, the triple point D 4 can have one or three real branches, etc. 11

12 Surfaces Consider a complex projective surface in P 3 C. Using modern intersection theory one finds formulas for the singular locus in terms of invariants of the desingularization of the surface. We saw an example in Pål s talk when the surface was the tangent developable of a space curve. (The subtle part is the use of residual intersection theory, when intersections are not of the expected dimension this was already dealt with in easy cases by the ancients.) 12

13 Classification of real surfaces in 3-space In addition to the case of quadrics, only cubics are fully classified including all possible singular types. There are several ways to do it, one way is via the number and types of real lines and the real types of eventual singular points, another via the Sylvester representation. The Sylvester representation yields a semialgebraic partition of an affine parameter space, where each region determines cubics of a given type. 13

14 Parameterized surfaces the Veronese (triangular) case The Veronese embedding of degree d is the map v d : P 2 P (d+2 2 ) 1, v d (t 0 ; t 1 ; t 2 ) = (t i 0 0 t i 1 1 t i 2 2 ) i0 +i 1 +i 2 =d Any rational surface in P 3 that can be parameterized by polynomials of degree d is a projection of the Veronese embedding. In the complex case a general projection has a double curve of degree m, t triple points, and ν 2 pinch points, given by m = d(d 1)(d 2 + d 3)/2 t = (d 6 12d 4 + 9d d 2 72d + 30)/6 ν 2 = 6(d 1) 2 14

15 The Steiner surface This is the case d = 2. The complex Steiner surface has a double curve of degree 3 and one triple point (which is also a triple point for the double curve). So the double curve consists of three lines meeting in one point. There are 6 pinch points, two on each of the three lines. There are 6 types of real Steiner surfaces. 15

16 Parameterized surfaces the Segre (tensor) case The Segre embedding of bidegree (a, b) is the map σ a,b : P 1 P 1 P (a+1)(b+1) 1, σ a,b ((s 0 ; s 1 ), (t 0 ; t 1 )) = (s i 0 0 s i 1 1 t j 0 0 t j 1 1 ) i0 +i 1 =a,j 0 +j 1 =b In the complex case a general projection has a double curve of degree m, t triple points, and ν 2 pinch points, given by m = a 2 b 2 2ab + (a + b)/2 t = 4ab(a 2 b )/3 8a 2 b 2 + 2ab(a + b) 8(a + b) + 4 ν 2 = 12ab 8(a + b)

17 The biquadric surface This is a projection of the Segre embedding with a = b = 2, and it has degree 8. The double curve has degree 10, there are 20 triple points and 20 pinch points. 17

18 The bicubic surface This is a projection of the Segre embedding with a = b = 3 and has degree 18. The double curve has degree 66, there are 520 triple points and 64 pinch points. Thus it is clear that this surface would not have been used to the extent that it is, if it were not for the fact that one usually only considers patches of this surface, where (almost) none of the singularities appear. Or, in other words, this surface is OK locally, though it is rather awful globally. So in what sense do we want to classify (the real part of) these surfaces?? Maybe we only want to look at them locally but then we might want to study singular patches too. 18

19 Suppose now that we project a Veronese embedding from a linear space that intersects the embedded surface in a certain number of points. For example, if we consider the case d = 3 and take as projection center the linear 5-space spanned by 6 points on the surface, we obtain a cubic surface. This cubic surface is equal to the blow-up of the plane in these six points. One way of classifying the cubic surfaces also in the real case is to look at the position of these 6 points in the plane. (The cubic surface is an example of a Del Pezzo surface for more on these, wait for the talk by Schicho. Note that the toric hexagonal surface mentioned by Krasauskas is also a special projection of v 3.) 19

20 Assume now that X is an implicitly defined cubic surface with a singular point, and that this point is P = (0, 0, 0, 1). Then the equation of the cubic is X 3 f 2 (X 0, X 1, X 2 ) + f 3 (X 0, X 1, X 2 ) = 0, and X is the blow-up of the points of intersection of f 2 = 0 and f 3 = 0. In fact, X has the parameterization (with base points) (X 0 ; X 1 ; X 2 ) (X 0 f 2 ; X 1 f 2 ; X 2 f 2 ; f 3 ). 20

21 Monoid surfaces A surface of degree d which is singular at the point P = (0; 0; 0; 1) with multiplicity d 1 has implicit equation of the form X 3 f d 1 (X 0, X 1, X 2 ) + f d (X 0, X 1, X 2 ) = 0, and a rational parameterization (X 0 ; X 1 ; X 2 ) (X 0 f d 1 ; X 1 f 2 ; X 2 f 2 ; f d 1 ). It is equal to the blow-up of the base points given by the intersection of the two curves defined by f d 1 = 0 and f d = 0. We are currently studying the case d = 4, hoping to find an interesting (sub)family that can be used for practical purposes. 21

22 Classification of (isolated) singularities There is an extensive theory for the classification of real and complex isolated hypersurface singularities, especially for plane curves and surfaces in 3-space. (Also for space curves, but this gets harder.) One associates certain invariants to the singularity, such as its Milnor and Tyurina numbers: the Milnor number is the dimension of the versal unfolding of the singularity, the Tyurina number is the dimension of the versal deformation space (some times these are equal). The Milnor number has a topological interpretation, but can be computed algebraically it is the intersection number of the partial derivatives of the defining equation at the point. A crude bound for the Milnor number of the singularity of a monoid degree d surface is therefore (by Bezout s theorem) (d 1) 3. 22