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Singularities of Some Projective Rational Surfaces

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Singularities of Some Projective Rational Surfaces Singularities of some projective rational surfaces Ragni Piene Centre of Mathematics for Applications & Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway [email protected] Abstract. We discuss the singularities of some rational algebraic sur- faces in complex projective space. In particular, we give formulas for the degrees of the various types of singular loci, in terms of invariants of the surface. These enumerative results can be used, on the one hand, to show the existence of singularities in the complex case, and, on the other hand, as an “upper bound” for the singularities that can occur on a real rational surface. 1 Introduction The simplest rational surfaces spanning projective 3-space are the quadric and cubic surfaces. The classification of real and complex quadric surfaces is very old, and the nonsingular and singular complex and real cubic surfaces were classified in the 19th century by Cayley, Salmon, Schl¨afli, Klein, Zeuthen (see [5], [13], [7], [19], [15]). Affine real 2- and 3-dimensional geometry was the object of intense study among 19th century mathematicians, but eventually the focus shifted to projective complex algebraic geometry, which from many points of view is much simpler. The classical work in algebraic geometry remains a rich source of inspiration for people working in Computer Aided Geometric Design (CAGD). In this article, the objects of study are surfaces that admit a rational pa- rameterization; because of the methods used, we choose to work in the complex projective setting. The two types of rational parameterizations most commonly used in CAGD are the “triangular” and “tensor product” parameterizations. These are patches of what in algebraic geometry are called Veronese and Segre surfaces. Some times these parameterizations have base points; we shall look at some examples, namely Del Pezzo surfaces and monoid surfaces. In the GAIA II project “Intersection algorithms for geometry based IT- applications using approximate algebraic methods” we realized the need for a better understanding of the singularities of algebraic curves and surfaces that are used in geometric modeling. Though the work presented here is carried out in the complex projective setting, the enumerative results can be interpreted as an “upper bound” for the singularities that may appear in the real affine case. There are of course a lot of hard and interesting problems that can not be addressed via these methods, such as determining the number of connected components and positions of a real curve or surface (cf. Hilbert’s 16th problem), questions of boundedness and of equidimensionality, and the problem of determining which real forms a given complex singularity may have. 2 Numerical characters of a projected surface N N Let PC , or P for short, denote the complex projective N-space. Let S be a nonsingular surface and let S,→ PN be any embedding of S. One can show that for most linear projections p: S → P3, the induced map p: S → X := p(S) is finite and birational, and X has only ordinary singularities. By this we mean the following: Let Γ := Sing X denote the singular locus of X, and let P ∈ Γ . Assume P = (1 : 0 : 0 : 0) and that f(x1, x2, x3) is the equation of X in the affine space where x0 6= 0. Consider the Taylor series expansion of f, which, by abuse of notation, we also denote by f. Then, up to change of coordinates, there are only three possibilities [9, p. 202]: 1. The point P is a point where the surface X intersects itself transversally, −1 i.e., we have f = x1x2+ terms of higher degree, and #p (P ) = 2. All but finitely many points of Γ are of this type. 2. The point P is a point where three sheets of the surface intersect transver- −1 sally, i.e., we have f = x1x2x3+ terms of higher degree, and #p (P ) = 3. This point is a triple point of the curve Γ and of the surface X. 3. The point P is a point of selfintersection of the surface where the projection 2 2 map is ramified. Here we have f = x2 − x1 x3+ terms of higher degree, and #p−1(P ) = 1. Such points are called pinch points. (The real part of the surface at such a point is the Whitney umbrella.) The curve Γ is called the nodal curve of X; its only singular points are the triple points (the pinch points are not singular on Γ ). Introduce the following numerical characters: n := deg X m := deg Γ t := # triple points of X and of Γ ν2 := # pinch points of X 2 2 c1 := KS = the self intersection of the canonical divisor of S c2 := the second Chern class of S (= the topological Euler characteristic of S). Then we have the following formulas [9, Prop. 1, p. 211]: 2 2 c1 = (n − 4) n − (3n − 16)m + 3t − ν2 (1) 2 c2 = n(n − 4n + 6) − (3n − 8)m + 3t − 2ν2 (2) Subtracting (1) from (2) gives 2 ν2 = 2n(2n − 5) − 8m + c1 − c2 (3) and substituting (3) in (1) gives 2 2 3t = 3(n − 8)m − n(n − 12n + 26) + 2c1 − c2 (4) The sectional genus π of S is the geometric genus of a general hyperplane section of S. By the adjunction formula it is given by 2π − 2 = deg KS + n. Among the other characters considered classically are: µ2 := class of X µ1 := rank of X 1 := rank of Γ Recall that the class of X is the number of tangent planes containing a given line, hence it is the same as the degree of the dual surface of X; the rank of X is the number of tangent lines in a given plane through a given point in that plane (hence is equal to the class of a plane section of X); the rank of the curve Γ is the number of tangent lines to Γ that meet a given line. From the formulas in [9, Thm. 2, p. 204; Prop. 3, p. 212] we deduce the following formulas for the characters of the projected surface: µ1 = n(n − 1) − 2m (5) µ2 = n(2n − 5) − 4m + c2 (6) 2 1 2 1 = n(n − 14n + 31) − 2m(n − 13) + 2 (3c2 − 5c1) (7) 3 The Veronese surfaces The dth Veronese embedding of the plane P2 is the map 2 N vd: P → P , d+2 where N := 2 − 1, given by i0 i1 i2 vd(t0 : t1 : t2) = (t0 t1 t2 )i0+i1+i2=d. In affine coordinates, this is just the triangular parameterization 2 2 d (t1, t2) 7→ (t1, t2, t1, t1t2, t2, . , t2) We want to study the surface X ⊂ P3 obtained by projecting the nonsingular 2 3 N surface S = vd(P ) to P from a linear subspace L ⊂ P of dimension N − 4. The surface X has a rational parameterization by polynomials of degree d, and, conversely, any surface in P3 that has such a parameterization is obtained in this way. 2 2 Assume d ≥ 2, L ∩ vd(P ) = ∅, and that the induced morphism p : P → X is birational. In this case, the degree of X is n = d2. Since the morphism p is finite (because it is a projection), it is equal to the normalization map of X. It is well known that therefore, X can have no isolated singular points, so that Sing X is either empty or purely one-dimensional. The following proposition shows that X must be singular, and gives a bound for the degree of the singular locus. Proposition 1. In the situation above, set Γ := Sing X. Then Γ is a curve, of degree 1 2 deg Γ ≤ 2 d(d − 1)(d + d − 3). Moreover, equality holds if and only if the components of Γ are ordinary double curves or cuspidal edges on X. Proof. Let H ⊂ P3 be a general plane, and set C := X ∩ H. Then C is a plane curve of the same degree, d2, as X. Let D denote the inverse image of C in P2; this is a plane curve of degree d, birationally equivalent to C. It follows from Bertini’s theorem that D is nonsingular, and therefore D is equal to the desingularization of C. Hence, the sectional genus π of S, by definition equal to the geometric genus g(C) of C, is equal to the geometric genus of D, namely d−1 π = g(C) = g(D) = 2 . By the genus formula for the plane curve C, we have d−1 d2−1 P 2 = 2 − x∈Sing C δx, where δx denotes the δ-invariant of the point x ∈ Sing C. Note that δx ≥ 1, with equality if and only if x is an ordinary double point (a node) or an ordinary cusp, i.e., if and only if the components of Γ are ordinary double curves or cuspidal P edges on X. Since d ≥ 2 by assumption, we get x∈Sing C δx > 0, hence C is singular. Using Bertini’s theorem again, we have Sing C = H ∩ Sing X = H ∩ Γ , therefore #Sing C = #H ∩ Γ = deg Γ . The last equality holds because, since H is general, we may assume that H and Γ intersect transversally. 2 If we consider a general projection of the dth Veronese surface S = vd(P ), 2 2 we have n = d , c1 = 9, and c2 = 3. Moreover, from Proposition 1, we have 1 2 m = 2 d(d − 1)(d + d − 3). Then (3) and (4) give the following formulas: 2 ν2 = 6(d − 1) 1 5 4 3 2 t = 6 (d − 1)(d + d − 11d − 2d + 42d − 30) From (5), (6), and (7), we get µ1 = 3d(d − 1) 2 µ2 = 3(d − 1) 2 1 = 3(d − 1) (d − 2)(d + 3) Example 1.
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