Counting projections of rational curves
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COUNTING PROJECTIONS OF RATIONAL CURVES
BY
Matteo Gallet∗ Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265, 34136 Trieste, Italy e-mail: [email protected]
AND
Josef Schicho∗∗ Research Institute for Symbolic Computation, Johannes Kepler University Altenbergerstraße 69, A-4040 Linz, Austria e-mail: [email protected]
ABSTRACT
Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree of the curves and the dimensions of the two given ambient projective spaces, the number of curves and projections fulfilling the requirements is finite. Using standard techniques in intersection theory and the Bott residue formula, we compute this number.
∗ Supported by the Austrian Science Fund (FWF): W1214-N15 Project DK9,
P26607, P25652, and Erwin Schr¨ odinger Fellowship J4253.
∗∗ Supported by the Austrian Science Fund (FWF): W1214-N15 Project DK9 and
P26607. Received August 7, 2017 and in revised form May 23, 2019
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M. GALLET AND J. SCHICHO
Isr. J. Math.
Introduction Inspired by problems in multiview geometry concerning image-object correspondence under projections (see for example [BKH13], and [HZ04] for a general account on the topic), we consider the following question, where all varieties are complex: given two general rational curves Ca ⊆ Pa and Cb ⊆ Pb , both of degree d ∈ N, and a natural number c ∈ N, find a rational curve Cc ⊆ Pc of degree d, together with two linear projections πa : Pc Pa , πb : Pc Pb such that πa (Cc ) = Ca and πb (Cc ) = Cb . We are interested in counting the number of such curves and projections, when this number is finite up to projective equivalence in Pc . Because of the rationality assumption, we can interpret the curves appearing in the previous formulation as images of maps fu : P1 −→ Pu for u ∈ {a, b, c}. In this way, we can translate the original problem into a problem of vector spaces of polynomials of degree d on P1 . Since we are only interested in the images of the maps fu , and not in the maps themselves, we need to allow possible reparametrizations, namely automorphisms of P1 . Once we apply this translation, the problem becomes: given two general vector subspaces Va , Vb ⊆ C[s, t]d of dimension a + 1 and b + 1, respectively, and a natural number c ∈ N, find automorphisms σ ∈ PGL(2, C) such that dim(Va + Vbσ ) ≤ c + 1. Here C[s, t]d is the vector space of homogeneous polynomials of degree d, and we denote by Vbσ the image of Vb under the action of σ, which operates on polynomials by applying the change of coordinates determined by σ to the variables. A dimension count shows that one may expect that if Va and Vb are general subspaces, and the condition (∗)
(a + b − c + 1)(d − c) = 3
holds, then there exists a finite number of automorphisms σ ∈ PGL(2, C) satisfying the requirements of the problem. In terms of the initial formulation, this means that if Equation (∗) holds and
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