Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 2, 173-180.

Simple Conditions on Conic Varieties

Konrad Drechsler and Ulrich Sterz

Fachbereich Mathematik und Informatik Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle/Saale drechsler@mathematik.uni-halle.d400.de

Abstract. Given a set of conditions B of codimension one on a conic variety obtained by successively blowing-up the P^{5} the conic varieties N will be described for which the intersection number of conditions computed in the intersection ring of N is geometrically significant. It will be proved that N is complete with respect to B if and only if N covers a suitable N^{0}. N^{0} will be determined, explicitely, in terms of the Newton-Cramer-polygon of a local representation of B.

1. Introduction

During the first part of our century Hilbert’s fifteenth problem [9] regarding the justification of Schubert’s enumerative calculus was solved in the sense that it was transformed from an enumerative problem into an intersection problem on a suitable smooth variety N. Intersection theory was then developed. To apply Schubert’s calculus is nothing else but working in the intersection ring of the variety N.

But from the start it is not evident on which variety N the enumerative problem can be adequately described as an intersection problem. In terms of the classical enumerative geometry, this means that a formula is either valid with respect to a “condition” B or not. Let Y be the support variety in which the enumerative problem is given. It is then understood that an element of a family of geometrical objects in Y satisfies a “condition” B if the point representing the element in N lies in a subvariety B of N. The fulfilment of a number of conditions equals its position in the intersection of the respective subvarieties. The intersection has to be proper.^{1} This can be accomplished by moving the subvariety B into an equivalent subvariety with respect to an equivalence relation ∼ belonging to the intersection theory. But this movement may destroy the geometric significance of B on the support variety Y.

In general, one has a canonical group action G × Y ⟶ Y on Y, which preserves the properties of B. It leads to an action G × N ⟶ N on N. The computations in the intersection ring of N give us geometrically significant numbers, if the intersection of each subset U of N with gB is proper for some element g of the group G. In [2] we called a variety N having this property G-complete with respect to B. As a result we simply need to make sure the intersection is proper for the G-stable subsets U of N.

1 Applying the modern intersection theory of Fulton or Vogel, one gets another approach to enumerative geometry [6], [12].

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1994 Heldermann Verlag, Berlin

If some G-stable U does not show this property, one way to complete the variety N is to blow it up in U. Having N complete it is a nontrivial problem to compute the intersection ring of N and to represent B in the ring.

The variety of conics is one of the important examples of classical enumerative geometry, which allows us to demonstrate the inherent problems and to solve them if possible. In previous notes [3] we constructed an infinite partially ordered set (N , ≺) of conic varieties N by successively blowing-up the five dimensional projective space which parameterizes the set of conics in the projective plane. We computed the intersection rings of all these varieties [4].

We were able to prove the following [3], [5]: There are varieties N of N GL(3)-complete with respect to B, where B is an element of a series of certain contact conditions of higher order. We were able to represent these conditions in the intersection ring of N. As a byproduct of these computations we proved, once again, Halphen’s second formula [8], [1].

In this paper we would like to extend these results to any simple conditions B; “simple” here means that the codimension of B in N is one. In this case G-completeness with respect to B is equivalent to the following: No G-stable subset of N is contained in B. We shall prove (Theorem 4) that N is G-complete with respect to B if and only if N^{0} ⪯ N for a suitable N^{0}. We determine N^{0}, explicitely, in terms of the Newton-Cramer-polygon of a local representation of B.

Thus computations concerned B in the intersection ring of N give us geometrically signi-ficant numbers if N^{0} ⪯ N. The multiplication of B with any cycle Γ of dimension one in this ring leads to the (total) intersection number of B and Γ . Representing B and Γ by suitable basises [4] one can get, for example, a formula known as Halphen’s 1^{st} formula for the “proper intersection number”of B and Γ [7]. A modern proof of this formula can be found in [1]. A generalization to higher dimensional quadrics was done in [10].

At the end of our paper we describe an N^{1} ∈ N with N^{1} ⪯ N^{0} so that the proper intersection number of B and Γ on N equals the total intersection number if N^{1} ⪯ N.

2. Conic varieties

Let M be any subset of Q_{+}^{∞} = Q_{+}^{∗} ∪ {0, ∞} where Q_{+}^{∗} = {q ∈ Q, q > 0} and let M̄ be the closure of M in the propped tree (Q_{+}^{∞}, ) described in [5], see Fig.1.

Define as in [3], [5] for each finite M̄ containing {0/1, 1/0} a conic variety N(M̄) by successive blowing-ups centered in the three dimensional GL(3)-stable subset Z_{M′}(a/b) of N(M̄′), a/b ∈ max(M̄), M̄′ = M̄ \ {a/b}. So we get a partially ordered set (N , ≺) of conic varieties N(M̄) by

N(M̄_{1}) ≺ N(M̄_{2}) ⇔ M̄_{1} ⊂ M̄_{2}.

We consider N(M̄) as a subvariety of the product of projective spaces Proj V(x/y), x/y ∈ M̄, over an algebraically closed field K of characteristic zero.

Let us start with the classical variety of “complete” conics N({0/1, 1/0}) contained in Proj V(1/0) × Proj V(0/1) where Proj V(1/0) represents the point conics and Proj V(0/1) represents the line conics. The stable subvarieties under the canonical action of GL(3) on N({0/1, 1/0}) are the four dimensional varieties E(0/1) and E(1/0) of the conics with the ranks (2, 1) and (1, 2) and the three dimensional Halphen variety Z(1/1) of the totally degenerated conics.

Generally, we have on N(M̄) the three dimensional GL(3)-stable subsets Z_{M}(u/v) for each u/v of the neighbourhood nbd(M̄) of M̄ in (Q_{+}^{∞}, ). The exceptional varieties E_{M}(a/b) of the blowing-ups centered in Z_{M′}(a/b) give the system of the four dimensional GL(3)-stable subsets E_{M}(x/y), x/y ∈ M̄. Let P_{M} and L_{M} be the divisors on N(M̄) defined by linear divisors on Proj V(1/0) and Proj V(0/1).

Fig.1: The propped tree thick lines - the tree relation thin lines - the prop relation

3. Completeness

Let B be a four dimensional subset of N({0/1, 1/0}). It is our aim in this section to determine the minimal M̄ for which N(M̄) is GL(3)-complete with respect to the proper image B_{M} of B on N(M̄). This is impossible if B ⊇ E(0/1) or B ⊇ E(1/0), but it is valid for M̄ = {0/1, 1/0} if Z(1/1) ⊄ B.

Suppose that B ⊉ E(0/1) and B ⊉ E(1/0) but Z(1/1) ⊂ B. Then in a suitable parameter representation R_{μ}^{ν}(1/1) with parameters α_{.}(1/1), α^{.}(1/1), γ, δ and ε [13], [5] the ideal of B can be generated by a polynomial

∑_{(i,j)∈A} K_{ij} · α^{.}(1/1)^{i} · α_{.}(1/1)^{j}  (1)

where K_{ij} ∈ K[γ, δ, ε]. A = A_{B} is the subset of all (i, j) ∈ N^{2} with K_{i,j} ≠ 0. The zero point (0, 0) is not in A, but there exist pairs of type (0, j) and (i, 0) in A with j ≠ 0 ≠ i.

Consider the Newton-Cramer set M = M_{B} of all elements x/y of Q_{+}^{∗} the opposite −x/y of which is the slope of a side of the Newton-Cramer polygon of (1). We denote this side by [x, y] thinking of it as a subset of N^{2}. Let r_{B}(x/y) + 1 be the number of points of [x/y].^{2}

Fix an element u/v ∈ nbd(M̄) and let x/y − u/v the lower neighbour of it. No element of M lies between x/y and (u − x)/(v − y) in (Q_{+}^{∞}, <) by Lemma 1.4 of [5]. Therefore, the sides of the Newton-Cramer polygon belonging to max_{<}{ξ/η ∈ M : ξ/η < u/v} and min_{<}{ξ/η ∈ M : ξ/η > u/v} have a common vertex (i_{0}, j_{0}) ∈ A if both exist. If, for example, max_{<}{ξ/η ∈ M : ξ/η < u/v} does not exist, we take (i_{0}, j_{0}) = (i, 0) ∈ A with minimal i.

Let R_{μ}^{ν}(u/v) be a parameter representation on N(M̄) with the parameters α_{.}(u/v), α^{.}(u/v), γ, δ and ε, [5], Lemma 1.5.

Lemma 1: The ideal in R_{μ}^{ν}(u/v) of the proper image B_{M} of B on N(M̄) is generated by the polynomial

∑_{(i,j)∈A} K_{ij} · α^{.}(u/v)^{x(i−i_{0})+y(j−j_{0})} · α_{.}(u/v)^{(u−x)(i−i_{0})+(v−y)(j−j_{0})}  (2)

if u/v < x/y and by

∑_{(i,j)∈A} K_{ij} · α^{.}(u/v)^{(u−x)(i−i_{0})+(v−y)(j−j_{0})} · α_{.}(u/v)^{x(i−i_{0})+y(j−j_{0})}  (3)

if u/v > x/y.

Proof: By a stepwize substitution parameters in (1) using

α_{.}(u/v) · α^{.}(u/v) = α^{.}(a/b), α_{.}(u/v) = α_{.}(a/b) if a/b < u/v  (4)

or

α_{.}(u/v) · α^{.}(u/v) = α_{.}(a/b), α^{.}(u/v) = α^{.}(a/b) if a/b > u/v,  (5)

(1.13) and (1.14) of [5], we get an element of the ideal of B_{M}:

∑_{(i,j)∈A} K_{ij} · α^{.}(u/v)^{xi+yj} · α_{.}(u/v)^{(u−x)i+(v−y)j}  (6)

if u/v < x/y and

∑_{(i,j)∈A} K_{ij} · α^{.}(u/v)^{(u−x)i+(v−y)j} · α_{.}(u/v)^{xi+yj}  (7)

2 The pair (x, y) is said in [1] to be a pair of local characteristic numbers of B and r_{B}(x/y)

its multiplicity. [7]

if u/v > x/y.

We can prove (6), for example, as follows: Let

M̄_{x/y} = {w/z ∈ M̄ : x/y ⊑ w/z}, M̄′ = M̄ \ M̄_{x/y} and x′/y′ − x/y.

Suppose that (6) and (7) hold with M̄′ instead of M̄ and x/y instead of u/v.

Since u/v < x/y we have to substitute by (5). If x′/y′ > x/y then (u − x)/(v − y) = (x−x′)/(y−y′). If x′/y′ < x/y we have (u−x)/(v−y) = x′/y′. In both cases we get by the induction hypothesis the polynomial (6).

The point (i_{0}, j_{0}) of A is choosen in such a way that all points of A lie above or on the lines l_{1} and l_{2} with the slopes −x/y and −(u − x)/(v − y) passing through (i_{0}, j_{0}). Therefore

x(i − i_{0}) + y(j − j_{0}) ≥ 0  (8)

and

(u − x)(i − i_{0}) + (v − y)(j − j_{0}) ≥ 0  (9)

for all (i, j) ∈ A.

In the case u/v < x/y we divide (6) by α^{.}(u/v)^{xi_{0}+yj_{0}} · α_{.}(u/v)^{(u−x)i_{0}+(v−y)j_{0}} to get (2). The equality holds in (8) and (9) at least at the point (i_{0}, j_{0}). Therefore, (2) contains no factor α^{.}(u/v) or α_{.}(u/v). Thus it generates the ideal of B_{M} because of (1.15), (1.16) in [5]. We prove the case x/y < u/v in a similar way.

Lemma 2: If M = M_{B} is the Newton-Cramer set of the equation of B in any parameter representation R_{μ}^{ν}(1/1) then N^{0} = N(M̄_{B}) is GL(3)-complete with respect to B_{M}.

Proof: Consider Z_{M}(u/v), u/v ∈ nbd(M̄). The polynomial (2) or (3) is not contained in the ideal (α^{.}(u/v) , α_{.}(u/v)) belonging to Z_{M}(u/v) in R_{μ}^{ν}(u/v) because the equality holds in (8) and (9) at least at the point (i_{0}, j_{0}) of A. Therefore, Z_{M}(u/v) is not contained in B_{M} for all u/v ∈ nbd(M̄). The variety E(x/y), x/y ∈ M̄, cannot lie in the proper image B_{M} of B. Thus, no GL(3)-stable subset of N(M̄) is in B_{M}. N(M̄) is GL(3)-complete with respect to B_{M}.

Lemma 3: Let M = M_{B} be the Newton-Cramer set of the equation of B in any parameter representation R_{μ}^{ν}(1/1) and a/b ∈ max(M̄), M′ = M̄ \{a/b}. Then N(M′) is not GL(3)- complete with respect to B_{M′}.

Proof: Let e/f ∈ Q_{+}^{∗} so that a/b − e/f and a/b > e/f, for example. Then min_{<}{ξ/η ∈ M : ξ/η > e/f} = a/b, and (i_{0}, j_{0}) is one of the vertices of [a/b]. Let (i_{1}, j_{1}) be the other. Then (i_{1}, j_{1}) = (i_{0} − rb, j_{0} + ra) where r = r(a/b). Put u/v = e/f and x/y = a/b in (2). Then the term of (2) for (i, j) = (i_{1}, j_{1}) is K_{i_{1}j_{1}} · α_{.}(e/f)^{r}. Therefore,

max_{(i,j)∈A}{(e − a)(i − i_{0}) + (f − b)(j − j_{0}) − a(i − i_{0}) − b((j − j_{0})} = m > 0.

Multiply (2) by α^{.}(e/f)^{m} and substitute as far as possible α_{.}(e/f) · α^{.}(e/f) = α_{.}(a/b) and α^{.}(e/f) = α^{.}(a/b). In this way we get a polynomial not divisible by α^{.}(a/b) or α_{.}(a/b). Therefore it generates the ideal of B_{M′} in R_{μ}^{ν}(a/b). But it does not have constant term. By Corollary 1.6 of [5] we get Z_{M′}(a/b) ⊂ B_{M′}.

Theorem 4: Let Z(1/1) ⊂ B ⊂ N({0/1, 1/0}), dim_{K} B = 4, E(0/1) ⊈ B and E(1/0) ⊈ B. Let M_{B} be the Newton-Cramer set of the equation of B in any parameter representation R_{μ}^{ν}(1/1) and T any finite subset of the tree (Q_{+}^{∞}, ). Then N(T̄) is GL(3)-complete with respect to B_{T} if and only if N^{0} = N(M̄_{B}) ⪯ N(T̄).

Proof: Let M̄ ⊆ T̄. Then N(T̄) is GL(3)-complete with respect to B_{T} if N(M̄) is GL(3)- complete with respect to B_{M} and this is true by Lemma 2. If M̄ ⊈ T̄ an element a/b ∈ M̄ exists which is in max(M̄) but not in T̄, and we continue as in Lemma 3.

In particular, we consider the subvariety B = S = S(x/y), x/y ∈ Q_{+}^{∗}, of N({0/1, 1/0}) which has the polynomial

(α^{.}(1/1))^{y} − (α_{.}(1/1))^{x}

for generator of its ideal. Then M_{S} consits of the single element x/y. Because we think of the elements of Q_{+}^{∗} to be reduced fractions, we have r_{S}(x/y) = 1 for all x/y ∈ Q_{+}^{∗}.

The conditions S(x/y) were introduced by Halphen. They can be described by cross ratios [8], [1].

If 1/1 x/y we gave S(x/y) a geometric interpretation in terms of a certain contact condition (see [5]) in generalization of Halphen’s example ([8], 3^{me} exemple).

S(1/1) could be changed by the condition of “superosculation” to a polar net of conics [11] or by the condition of “mediate superosculation” to a conic (π̄, λ̄) as it was done in [5]. Then r_{S}(1/1) = 1, too, because the generating polynomial is

\begin{matrix}ᾱ_{.}α^{.}[α_{.}α^{.} + α_{.}(δ − δ̄)^{2} + ((ε − ε̄) − (γ − γ̄)δ̄)^{2} + ᾱ^{.}(α_{.} + (γ − γ̄)^{2} + ᾱ_{.}ᾱ^{.}]^{3} \\ − α_{.}ᾱ^{.}[ᾱ_{.}ᾱ^{.} + ᾱ_{.}(δ̄ − δ)^{2} + ((ε̄− ε) − (γ̄ − γ)δ)^{2} + α^{.}(ᾱ_{.} + (γ̄ − γ)^{2}) + α_{.}α^{.}]^{3}, \end{matrix}

where α_{.}(1/1) = α_{.}, α^{.}(1/1) = α^{.} and ᾱ^{.}, ᾱ_{.}, γ̄, δ̄, ε̄ are the parameters of (π̄, λ̄).

Corollary 5: N(M̄) is GL(3) – complete with respect to S_{M}(x/y) for all x/y ∈ M̄ \ {0/1, 1/0}.

Let S_{M}(0/1) = 2L_{M}, S_{M}(1/0) = 2P_{M}.

In [4] we described the intersection rings H^{∗}N(M̄) over Q of all N(M̄) ∈ N as graded rings in terms of factor rings of polynomial rings.

Theorem 4.3 of [4] says that the S_{M}(x/y), x/y ∈ M̄, form a basis of H^{∗}N(M̄).

4. Remark

Let

B ∼ αP + βL in H^{1}N({0/1, 1/0})

and

B_{M} ∼ ∑_{x/y∈M̄} b(x/y) · S_{M}(x/y) in H^{1}N(M̄).

The multiplication of the last relation by E_{M}(ξ/η)P^{2}L for all ξ/η ∈ M̄ \ {0/1, 1/0} gives the coefficients in the basis representation of B_{M} in H^{∗}N(M̄) described by Chasles’ characteristic numbers α and β and by the multiplicities r_{B}(x/y) :

\begin{aligned}b(x/y) & = \{ \begin{matrix}r_{B}(x/y) & if x/y ∈ M, \\ 0 & if x/y ∈ M̄^{∗} \ M, \end{matrix} \\ b(0/1) & = \frac{β}{2} − ∑_{x/y∈M} y · r_{B}(x/y), b(1/0) = \frac{α}{2} − ∑_{x/y∈M} x · r_{B}(x/y). \end{aligned}

Let Γ be a curve on N({0/1, 1/0}) no component of which lies in E(0/1) or E(1/0).

Define the proper intersection number of B_{M} and Γ_{M} on N(M̄) in an equivalent way as it is done on N({0/1, 1/0}) in [1], p. 25.^{3} The proper number does not change under the blowing-ups considered in 2. This can be proved by use of the substitution parameters (4) or (5) in the polynomial of B and in a representation of a branch of Γ.

The proper intersection number of B_{M} and Γ_{M} equals the total intersection number if M̄ ⊇ M̄_{B} because no point of N(M̄) exists so that the orbit of it lies in B_{M}. Thus, for example, Halphen’s 1^{st} formula about the proper intersection number [Ha1, CX] can be understood for all conic varieties N with N^{0} ⪯ N as an element of H^{5}N represented by the basis {S(x/y)}_{x/y∈M̄} and the basis {E(x/y)}_{x/y∈M̄} of H^{∗}N considered in [4], Theorem 4.3 and Lemma 3.1.

However, to get the fact that the proper intersection number of B_{M} and Γ_{M} equals the total intersection number M̄_{B} can be replaced by M̄_{B} ∩ M̄_{Γ} where M_{Γ} is the union of the pairs of the characteristics of the branches of Γ with the support in Γ ∩ Z(1/1). It may N^{1} = N(M̄_{B} ∩ M̄_{Γ}) ≺ N^{0}, possibly.

References

[1] Casas-Alvero, E., and S. Xambó-Descamps: The Enumerative Theory of Conics after Halphen. Lecture Notes in Mathematics, 11960, Springer - Verlag, Berlin - New York, 1986.

[2] Drechsler, K., and U. Sterz: On Completeness of Varieties in Enumerative Geometry. Beitr. Algebra Geom. 26 (1987), 53-71.

[3] Drechsler, K., and U. Sterz: Completion of the Variety of Conics with respect to Contact Conditions II. Beitr. Algebra Geom. 30 (1990), 117-132.

[4] Drechsler, K., and U. Sterz: The Intersection Rings of the Conic Varieties. Beitr. Algebra Geom. 31 (1990), 143-161.

[5] Drechsler, K., and U. Sterz: Contact Conditions on Conic Varieties and Halphen’s 2^{nd} Formula. Beitr. Algebra Geom. 33 (1992), 55-69.

3 Halphen [7] thought of it as the right number of conics common B and Γ.

[6] Fulton, W., and R. MacPherson: Defining Algebraic Intersections. Lecture Notes in Math. 687, Springer, Berlin 1978, 1-30.

[7] Halphen, G.-H.: Sur la théorie des caractéristiques pour les coniques. Proc. Lond. Math. Soc. 9 (1877-78), 149-177. Repr. in Math. Ann. 15, (1879), 16-44.

[8] Halphen, G.-H.: Sur le nombre des coniques qui, dans un plan, satisfont à cinq conditions projectives et indépendantes entre elles. Proc. Lond. Math. Soc., 1. Ser. 10, (1879), 76-87.

[9] Hilbert, D.: Mathematische Probleme. Gött. Nachr. 1900, 253-297.

[10] Procesi, C., and S. Xambó-Descamps: On Halphen’s first formula. Enumerative Algebraic Geometry (Copenhagen, 1989), 199-211. Contemp. Math. 123, Amer. Math. Soc., Providence, 1991.

[11] Semple, J. G.: Note on Halphen conditions. J. Lond. Math. Soc. 26 (1951), 122-125.

[12] Stückrad, J. and W. Vogel: An Algebraic Approach to the Intersection Theory. The Curves Seminars at Queen’s, II, Kingston, Canada, 1982, 1-32.

[13] Van der Waerden, B. L.: Zur algebraischen Geometrie XV. Lösung des Charakteristikenproblems für Kegelschnitte. Math. Ann. 115 (1938), 645-655.

Received May 19, 1993; revised version April 22, 1994.