VERONESE SURFACES AND LINE ARRANGEMENTS

TOM GRABER AND KRISTIAN RANESTAD^{∗}

Abstract. We formulate a generalization of Castelnuovo‘s lemma on rational normal curves to d-uple Veronese surfaces and apply it to define a Hilbert scheme compactification of line arrangements in the Hilbert scheme of Veronese surfaces. In the case of arrangements of five lines, we find the compactification to be isomorphic to a Del Pezzo surface of degree 5. We thus recover previous results of Kapranov and Hacking.

Paul Hacking recently studied compact moduli of hyperplane arrangements using stable pairs of a possibly reducible variety and a divisor on it [1]. His compactification coincides with a Hilbert scheme compactification of so called special Veronese varieties due to Kapranov [2].

In this note we consider line arrangements in the plane and define what coincides with Kapranov’s special Veronese varieties of dimension 2. In particular, we recover an analogue of Castelnuovo’s lemma on rational normal curves for d−uple Veronese surfaces. This sets up a natural map of Hacking’s moduli space of line arrangements of degree d + 3 to the Hilbert scheme of d−uple Veronese surfaces. We give a different and direct proof that the corresponding Hilbert scheme compactification of the moduli of stable 5-line arrangements is naturally isomorphic to a smooth Del Pezzo surface of degree 5. Our enumeration of singular surfaces in the family, recovers Hacking’s classification.

1. A Castelnuovo type lemma

We start with an analogue of

Lemma 1.1. [Castelnuovo] Through d + 3 points in linear general position in ℙ^{d} there is a unique rational normal curve.

due to Kapranov:

Lemma 1.2. [2] Given d+4 general ℙ^{d}’s in P^{N} with N = (d+2)(d+1)/2−1, such that any two of the ℙ^{d}’s meet in exactly a point, and any set of d + 1 intersection points span a ℙ^{d}. Then there is a unique d-uple Veronese surface that contains the (d + 4)(d + 3)/2 intersection points.

Remark 1.3. Note that we need the first “general”, since our proof is not constructive.

Proof. First we assume that V is a d−uple Veronese surface through all the intersection points. By Castelnuovo’s lemma, in each ℙ^{d} there is a unique rational normal curve of degree d through the d + 3 intersection points. We show that V actually must intersect each ℙ^{d} in the unique rational normal curve given by Castelnuovo’s lemma. For this, fix a ℙ^{d} and let Γ be the d + 3 intersection points in it. Let C be a general curve in the linear system |(d + 3)L| on V that contains Γ, where L is the class of a line in V ≅ ℙ^{2}. Since Γ is smooth, C is smooth. In fact d + 3 lines through the d + 3 points generate a linear system without base points outside Γ, so by Bertini C is smooth. Since C ⊂ V is the canonical embedding, Γ moves in a net on C. But C has only one g_{d+3}^{2}, so Γ must therefore coincide with the L ∩ C for some rational normal curve L_{0} ∈ |L| of degree d on V. Clearly L_{0} coincides with the unique rational normal curve given by Castelnuovo’s lemma. Thus V contains d + 4 rational normal curves of degree d.

Date: December 10, 2003.

1991 Mathematics Subject Classification. Primary 14J; Secondary 14D.

∗ Supported by the Research Council of Norway and by MSRI.

On the other hand any set of d + 4 general members of the linear system |L| on V gives rise to d + 4 pairwise intersecting ℙ^{d}’s in ℙ^{N}.

Now, consider the family of marked Veronese surfaces, where the marking is a line arrangement of degree d + 4, i.e. a collection of d + 4 lines in the plane. Clearly this family is irreducible. We compute the dimension by adding the dimension of the orbit of Veronese surfaces to the dimension of the family of line arrangements and get

(d + 2)^{2}(d + 1)^{2}/4 − 9 + 2(d + 4) = d(d + 4)(d^{2} + 2d + 5)/4

Compare this with the family of sets of d+4 pairwise intersecting ℙ^{d}’s. This family is clearly also irreducible. We compute its dimension as a configuration of d = 4 points in the Grassmannian satisfying the relevant Schubert condition and get

(d+4)(d+1)((d+2)(d+1)/2−(d+1))−((d+2)(d+1)/2−2(d+1)+1)(1+2+...+d+3)

= d(d + 4)((d + 1)^{2}/2 − (d − 1)(d + 3)/4) = d(d + 4)(d^{2} + 2d + 5)/4.

Since the two families have the same dimension, the lemma follows as soon as the Veronese surface is unique for a general configuration of d-planes. For this, let C be the line arrangement of degree d + 4. The d-th Veronese embedding is given by K_{C} − L_{C}, where L_{C} is the divisor of the embedding in ℙ^{2}. The divisor L_{C} is unique (given by the divisors determined by its d+4 line components), so K_{C} − L_{C} is unique, and therefore also the Veronese surface. ⊓ ⊔

2. Compactifying the muduli of stable line arrangements

Consider the family Y_{d} of d−uple Veronese surfaces that intersect d + 3 pairwise intersection ℙ^{d}’s in a rational normal curve. Now, the family X_{d} of ℙ^{d}’s that intersect each of the d + 3 fixed ℙ^{d}’s has dimension 2d in the Grassmannian.

Note that, by Lemma 1.2, there is a rational map X_{d} → Y_{d} whose general fiber is a (dual) Veronese surface. Thus

Proposition 2.1. The family Y_{d} of d−uple Veronese surfaces that intersect d + 3 pairwise intersecting ℙ^{d}’s in a rational normal curve, has dimension 2d − 2.

Definition 2.2. A line arrangement of degree d > 3 is called stable, if no three lines are concurrent.

The family of stable line arrangements of degree d + 3 modulo automorphisms has dimension 2d − 2. Now, clearly any Veronese surface in Y_{d} defines a stable line arrangement of degree d + 3.

Lemma 2.3. Two Veronese surfaces in Y_{d} determine distinct stable line arrangements of degree d + 3.

Proof. The embedding of a line arrangement of degree d + 3 on a d-uple Veronese surface is given by the canonical linear series, so the Veronese surface is unique.⊓ ⊔

Therefore the moduli of stable line arrangements has a compactification in the Hilbert scheme of d-uple Veronese surfaces. More precisely, we define Y_{d}^{H} to be the reduced closure of Y_{d} in the Hilbert scheme of Veronese surfaces.

Here we treat the case of d = 2. So we consider the family of Veronese surfaces in ℙ^{5}, defined according to the lemma by fixing five ℙ^{2}’s and letting a sixth plane P move in the family of planes that meet all the first five. This family of planes form a subvariety X in Gr(3, 6) of dimension 4, since it is the proper intersection of five Schubert hyperplane sections, while the corresponding family Y_{2} of Veronese surfaces has dimension 2.

Let us show directly how a stable line arrangement of degree 5 gives a Veronese surface intersecting five given pairwise intersecting planes in conics. So, consider a stable line arrangement given by linear forms l_{0}, l_{1}, … l_{4} on ℙ^{2}. In terms of coordinates in ℙ^{2} we may assume that l_{1} = y_{0}, l_{2} = y_{1}, l_{3} = y_{2}, l_{4} = y_{0} + y_{1} + y_{2} and l_{5} = y_{0} + e_{1}y_{1} + e_{2}y_{2}, where e_{1}e_{2} ≠ 0, e_{1} ≠ 1, e_{2} ≠ 1 and e_{1} ≠ e_{2}. Given five planes P_{i} in ℙ^{5} that pairwise intersect, we may choose coordinates such that the first four planes P_{1}, …, P_{4} are defined by x_{0} = x_{1} = x_{2} = 0, x_{2} = x_{3} = x_{4} = 0, x_{0} = x_{4} = x_{5} = 0 and x_{1} = x_{3} = x_{5} = 0, and that the fifth plane is defined by

x_{0} + a_{3}x_{3} + a_{4}x_{4} + a_{5}x_{5} = 0,
x_{1} + b_{3}x_{3} + b_{4}x_{4} + b_{5}x_{5} = 0,
x_{2} + c_{3}x_{3} + c_{4}x_{4} + c_{5}x_{5} = 0,

where

det (\begin{matrix}a_{3} & a_{4} & a_{5} \\ b_{3} & b_{4} & b_{5} \\ c_{3} & c_{4} & c_{5} \end{matrix}) = 0, c_{5} = 0, b_{4} = 0 and a_{3} = 0,

to make the fifth plane intersect the first four. This reduces to the condition that the equations of the fifth plane P_{5} are

x_{0} + a_{4}x_{4} + a_{5}x_{5} = 0,
x_{1} + b_{3}x_{3} + b_{5}x_{5} = 0,
x_{2} + c_{3}x_{3} + c_{4}x_{4} = 0,

and

det (\begin{matrix}0 & a_{4} & a_{5} \\ b_{3} & 0 & b_{5} \\ c_{3} & c_{4} & 0 \end{matrix}) = 0.

The last condition may be reformulated to say that there exist nonzero coefficients s_{1}, s_{2}, s_{3} such that −s_{2}b_{3} + s_{3}c_{3} = s_{1}a_{4} + s_{3}c_{4} = s_{1}a_{5} − s_{2}b_{5} = 0. Thus we may give the equations of P_{5} the form:

s_{1}x_{0} + Bx_{4} + Cx_{5} = 0,
s_{2}x_{1} + Ax_{3} + Cx_{5} = 0,
s_{3}x_{2} + Ax_{3} − Bx_{4} = 0.

We now define the Veronese map by a choice of basis for the quadrics on ℙ^{2}:

d_{0}x_{0} = l_{1}l_{3}, d_{1}x_{1} = l_{1}l_{4}, d_{2}x_{2} = l_{1}l_{2}, d_{3}x_{3} = l_{2}l_{4}, d_{4}x_{4} = l_{2}l_{3}, d_{5}x_{5} = l_{3}l_{4}

where the coefficients d_{0}, …, d_{5} are all nonzero and to be determined. In coordinates y_{i} we get

d_{0}x_{0} = y_{0}y_{2}, d_{1}x_{1} = y_{0}(y_{0} + y_{1} + y_{2}), d_{2}x_{2} = y_{0}y_{1},
d_{3}x_{3} = y_{1}(y_{0} + y_{1} + y_{2}), d_{4}x_{4} = y_{1}y_{2}, d_{5}x_{5} = y_{2}(y_{0} + y_{1} + y_{2}).

The image of l_{5} = 0 lies in the plane defined by the three linear forms

e_{2}d_{0}x_{0} + (d_{1}x_{1} − d_{0}x_{0} − d_{2}x_{2}) + e_{1}d_{2}x_{2},
d_{2}x_{2} + e_{2}d_{4}x_{4} + e_{1}(d_{3}x_{3} − d_{2}x_{2} − d_{4}x_{4}),

and

d_{0}x_{0} + e_{1}d_{4}x_{4} + e_{2}(d_{5}x_{5} − d_{0}x_{0} − d_{4}x_{4})

or after reordering and reduction

(1 − e_{2})d_{0}x_{0} + (e_{1} − e_{2})d_{4}x_{4} + e_{2}d_{5}x_{5}

d_{1}x_{1} + e_{1}d_{3}x_{3} + e_{2}d_{5}x_{5},

and

(1 − e_{1})d_{2}x_{2} + e_{1}d_{3}x_{3} + (e_{2} − e_{1})d_{4}x_{4}.

So this plane coincides with P_{5} when

(1 − e_{2})d_{0} = s_{1}, d_{1} = s_{2}, (1 − e_{1})d_{2} = s_{3}, e_{1}d_{3} = A, (e_{1} − e_{2})d_{4} = B, e_{2}d_{5} = C, which determines the coefficients d_{0}, …, d_{5} as promised. An immediate consequence of this computation is

Proposition 2.4. The surface Y is rational.

We shall now describe a compactification Y of Y_{2}, that we shall eventually show coincides with the Hilbert scheme compactification Y_{2}^{H}. For each Veronese surface the union of secant lines form a singular cubic hypersurface. The cubic hypersurface is defined by the determinant of a symmetric 3 × 3 matrix of linear forms, and it is singular precisely along the Veronese surfaces. Therefore Y_{2} as a reduced variety has a natural embedding in the space of cubic hypersurfaces. We take Y to be the reduced closure of Y_{2} in this embedding. Our main result in this section is the description of Y.

Theorem 2.5. The compactification Y of the family Y_{2} of Veronese surfaces in ℙ^{5} that meet five pairwise intersecting planes in conics, form a smooth Del Pezzo surface of degree 5. The lines on the Del Pezzo surface form the boundary Y \ Y_{2} and parameterizes degenerate Veronese surfaces. A point that lies on one line corresponds to a surface with a smooth cubic scroll component and a plane through the directrix of the scroll, while the intersection point between two lines corresponds to a surface with a smooth quadric component and two plane components meeting each other in a point and the quadric in a line, one from each ruling.

Remark 2.6. Kapranov [2] proves that by association, the moduli space Y_{2} is isomorphic to the moduli space M_{0,5} of 5-pointed ℙ^{1} which is isomorphic to a Del Pezzo surface of degree 5.

Proof. Consider the family of cubic hypersurfaces singular along a Veronese surface of the family Y_{2}. Their linear span is contained in the vector space V of cubic hypersurfaces that are singular at the ten points of intersection of the five planes P_{i}. This space V has dimension at least 6: The space of cubics that contain the five pairwise intersecting planes is 16-dimensional. A singularity at an intersection point imposes one linear condition, so altogether the singularities imposes at most ten conditions, which leaves at least a 6-dimensional vector space for the 10-nodal cubics.

Lemma 2.7. No cubic in V is singular along one of the five planes P_{i}.

Proof. Let F be a cubic singular along the plane P_{1}, say. If so let P′ be a 3−space spanned by P_{1} and the point of intersection between two of the other planes. Since F is singular in the intersection point, the span P′ must be contained in F. Consider the hyperplane H_{12} spanned by P_{1} and P_{2}. By the previous argument the restriction of F to H_{12} decomposes into three ℙ^{3}’s that contain P_{1}, one for each intersection point of P_{2} with the remaining three planes. On the other hand F contains P_{2}, so H_{12} must be a component of F. Similarly the hyperplanes spanned by P_{1} and P_{3}, P_{4} and P_{5} are all components of F. But F is a cubic, so this is absurd. ⊓ ⊔

Consequently, there is precisely a net of cubics in V that are singular at some point in P_{i} outside the original four. In fact since the cubics already contain P_{i}, it is at most 3 conditions to be singular at some point in the plane, and if it is less than 6 conditions to be singular at two general points outside the original four, then there would be a cubic hypersurface singular along the whole plane, contradicting Lemma 2.7. In particular

Corollary 2.8. The space ℙ(V) of cubic hypersurfaces that are singular in the ten intersection points of the five planes P_{i} is 5−dimensional and the cubic hypersurfaces that are singular at some point in P_{i} outside the original four form a ℙ^{2} and are all singular along the unique conic passing through the five points.

Let V_{i} ⊂ ℙ(V) be the subvariety of cubic hypersurfaces that are singular along some conic in P_{i}.

Lemma 2.9. The variety V_{i} is a rational cubic scroll, the Segre embedding of ℙ^{1} × ℙ^{2}.

Proof. By Corollary 2.8 there is a plane in V_{i} of cubic hypersurfaces for each conic. Since the conics moves in a pencil, the variety V_{i} is the union of a pencil of planes. Furthermore, no two planes in this pencil have a common point, because the corresponding cubic hypersurface would be singular along P_{i} contradicting Lemma 2.7. But the only pencils of planes with this property are those of the Segre embedding of ℙ^{1} × ℙ^{2}, so V_{i} must be such a cubic scroll. ⊓ ⊔

The five cubic scrolls V_{i} are distinct: Let H_{ij} be the hyperplane spanned by P_{i} an P_{j}, let p_{ij} be the intersection point P_{i} ∩ P_{j}, and let H_{i}(jk, lm) be the hyperplane spanned by P_{i} and the points p_{jk} and p_{lm}. Then H_{25} ∪ H_{34} ∪ H_{5}(12, 34) and H_{12} ∪ H_{34} ∪ H_{1}(25, 34) form cubic hypersurfaces that belong to ℙ(V). They are both singular along the conic (H_{25} ∪ H_{34}) ∩ P_{1} in P_{1}, but they are singular at the distinct conics (H_{25} ∪ H_{13}) ∩ P_{4} and (H_{35} ∪ H_{12}) ∩ P_{4} in P_{4}. Since the planes in V_{i} correspond to conics in P_{i}, the planes in V_{1} and V_{4} say, cannot coincide. Therefore, the V_{i} are also distinct.

Now, a cubic scroll is defined by the 2 × 2 minors of a 2 × 3 matrix of linear forms, i.e. be three quadrics. Since the intersection of two of these quadrics is a threefold of degree 4, two distinct cubic scrolls have at most one defining quadrics in common. Therefore the ideal of Y contains at least 5 quadrics, and Y is contained in five distinct cubic scrolls. In particular, Y has degree at most 5.

Next, we will enumerate degenerations of Veronese surfaces in V_{2}. So we consider limit schemes T in the Hilbert scheme of the Veronese surfaces in our family. A limit scheme T must clearly satisfy the following two conditions:

1. The Hilbert polynomial of T coincides with that of a smooth Veronese surface. In particular it has a surface component of degree 4 whose reduced part is connected in codimension 1.

2. The scheme theoretic intersection with any one of the five planes is at least a plane conic that passes through the four intersection points of this plane with the other planes.

First, we need some notation. As above we denote by p_{ij} the intersection point P_{i} ∩ P_{j}, and by H_{ij} the hyperplane spanned by P_{i} and P_{j}. Denote by P_{ijk} the plane spanned by the interesection points of P_{i}, P_{j} and P_{k}, and let L_{ijk} be the line of intersection P_{ijk} ∩ H_{lm}.

To enumerate the possible degenerate Veronese surfaces, we first exclude two cases. First, one of the planes P_{i} cannot be a component of a degenerate Veronese surface: If so, the residual surface S has degree 3 and intersect the four remaining planes in at least a conic. Any two of these four planes span a hyperplane that intersect S in a curve of degree at least 4, so S has a component in each of these 6 hyperplanes. But any three of these hyperplanes intersect in a plane, therefore S has at least 4 components. This is a contradiction. Secondly, any quadric surface component cannot intersect one of the five planes, say P_{5} in a conic: If so this component intersects the remaining four planes in at most two lines, in which case these lines meet at an intersection point of two planes, say p_{34}. The residual surface S has degree 2 and intersect P_{1} and P_{2} in at least a conic, and P_{3} and P_{4} in at least a line. Again, this means that the hyperplanes spanned by H_{12}, H_{13}, H_{14}, H_{23} and H_{24} all contain a component of S. This is possible only if S consists of the two planes P_{123} and P_{124}. But then the intersection of S and the quadric is 0-dimensional, against condition 1 above.

Now, any component of a degenerate Veronese surface meets a plane P_{i} in a conic section, a line, a point or not at all. Consider components of reducible degenerate Veronese surfaces. By the first condition, they are planes, quadrics or cubic scrolls. Since at most two planes P_{i} lie in a ℙ^{4} any cubic scroll intersects the union of the P_{i} in a curve of degree at most 7. In that case it intersect two planes in a conic, and the remaining three in a line. Furthermore these three lines are disjoint, so the scroll is smooth. Since a ℙ^{3} intersect at most 4 planes in a line, and no quadric component can intersect a plane in a conic, the intersection between a quadric component and the five planes is at most 4 lines. In that case they form a quadrangle defined by the pairwise intersection of two pairs of planes and the quadric surface component is smooth. Finally, any plane intersect the five planes in at most three lines, in which case it is the plane spanned by the intersection points of three planes.

Lemma 2.10. If T is a singular Veronese surface in the family Y, then T is the reducible union of a plane and a smooth cubic scroll intersecting the plane along the directrix, or the reducible union of two planes and a smooth quadric surface intersecting the planes along a line one from each ruling.

Proof. First, an irreducible singular degeneration of a Veronese surface is a cone over a rational normal curve. Since the intersection with each plane is a conic, the vertex of the cone is in each plane, absurd. A nonreduced component is a multiple structure on a plane or a double structure on a quadric. The bound of the intersection of the reduced component with the five planes excludes this possibility also. Therefore a singular Veronese surface in the family is reduced, and the components are planes, smooth quadrics or cubic scrolls. Furthermore, by degree reasons alone, any plane component must meet the union of the five planes in 3 lines, any quadric surface component is smooth and must meet the five planes in 4 lines. Furthermore if the singular Veronese surface has a quadric surface component, the residual components must be two planes, that meet the quadric in a line, one from each ruling. Finally, any cubic scroll intersect the five planes in a curve of degree 7 with two conic components and 3 disjoint lines. In particular the scroll is smooth, and the residual component is a plane through the directrix. ⊓ ⊔

We may now enumerate the degenerate Veronese surfaces in the above notation:

1. Any plane component coincides with the plane spanned by the intersection points of three planes P_{ijk} =< p_{ij}, p_{jk}, p_{ik} >.

2. Any quadric component is smooth and coincides with the quadric through say L_{123} and L_{345} which contain the points p_{14}, p_{24}, p_{15}, p_{25}. This quadric lies in the 3-space < P_{1}, P_{2} > ∩ < P_{4}, P_{5} >.

3. Any cubic scroll component is smooth and coincides with a scroll through three lines and meeting two planes in conics through their intersection point. For each pair of planes there is a pencil of them, i.e. for each plane component there is a pencil of them. For the plane P_{ijk} this pencil of cubic scrolls have the line L_{ijk} as a common directrix.

Altogether we find ten plane components in degenerations, fifteen smooth quadric surface components of degenerations, and ten pencils of smooth cubic scroll components.

For each of these degenerate Veronese surfaces the union of secant lines form a cubic fourfold: If the Veronese surface is smooth, then the cubic is the determinant of a symmetric matrix. If the surface is the union of a plane and a cubic scroll, then the cubic is the union of the ℙ^{4} of the cubic scroll and the unique rank 3 quadric with vertex the plane which contains the cubic scroll. If the surface is the union of two planes and a quadric surface, then the cubic is the union of three ℙ^{4}’s. Notice furthermore that in each case the union of the secant lines form the unique cubic hypersurface that is singular along the degenerate Veronese surface.

Therefore there is a curve consisting of ten lines in ℙ(V) parameterising singular cubics of degenerate Veronese surfaces.

Lemma 2.11. The surface Y ⊂ ℙ(V) contains at most ten lines.

Proof. Assume that a line in Y contains the cubic of a smooth Veronese surface, and consider the base locus of the pencil. Since every cubic of the pencil is singular along some Veronese surface, by Bertini, the union of the Veronese surfaces must be contained in the base locus of the pencil. In at least one of the planes P_{i} the pencil of Veronese surfaces restricts to the pencil of conics through the four intersection points, since otherwise the pencil of Veronese surfaces have five conics in common, which is impossible. Corresponding to the singular conics in the pencils, the Veronese must be degenerate, so the cubic is decompose in a hyperplane and a rank 3 quadric with vertex a plane P_{ijk}, or in three hyperplanes. If the base locus contains a Veronese surface, this must be contained in the rank 3 quadric and the vertex plane is a tangent plane to the Veronese surface. For each singular conic there is a rank 3 quadric containing the Veronese surface, such that the vertex plane is tangent to the Veronese surface. But two of these three quadrics must have a vertex plane that passes through the same intersection point p_{ij} on P_{i}. On the other hand this is a point on the Veronese surface, so in fact it must be the point of tangency, and the two vertex planes must coincide, which is absurd. Therefore all cubics of the pencil are cubics of degenerate Veronese surfaces, and the ten lines of degenerate Veronese surfaces are the only possible lines on Y. ⊓ ⊔

Clearly, both Y and the ten lines of degenerate Veronese surfaces lie in the five cubic scrolls V_{i}. Recall that Y has degree at most 5. On the other hand Y spans ℙ(V): Otherwise it is a possibly reducible cubic scroll, but Y contains contains at most ten lines. If Y has degree 4, it is again a scroll or a possibly degenerate Veronese surface. Only in the case of a Veronese surface are there at most ten lines. But a Veronese surface is not contained in any smooth cubic threefold scroll, so we are left with the sole possibility that Y has degree 5 and is contained in 5 quadrics. This is possible only if the ideal of Y is generated by five quadrics, the Pfaffians of a 5 × 5 matrix of linear forms. Thus Y is the scheme theoretic intersection of the cubic scrolls, the ten lines all lie on Y and Y is a smooth Del Pezzo surface. ⊓ ⊔

Corollary 2.12. Y coincides with the Hilbert scheme compactification Y_{2}^{H} of Y_{2}.

Proof. The family Y of surfaces is clearly flat, so there is a morphism f : Y → Y_{2}^{H}. On the other hand the singular cubic hypersurfaces associated to each surface in Y sets up a morphism g : Y_{2}^{H} → Y inverse to f, so f must be an isomorphism. ⊓ ⊔

Acknowledgement. We thank Paul Hacking for pointing out Kapranov’s work on special Veronese varieties, and Andreas Holmsen for posing a question on line transversals to certain sets of planes in ℙ^{4} that inspired us to consider these families of Veronese surfaces.

References

[1] P. Hacking: Compact moduli of hyperplane arrangements, preprint, math.AG/0310479

[2] Kapranov, M. M. Veronese curves and Grothendieck-Knudsen moduli space \overline{M}_{0,n}. J. Algebraic Geom. 2 (1993), no. 2, 239–262.

Department of Mathematics, University of California, Berkeley, CA 94720

   E-mail address: graber@math.berkeley.edu

Matematisk Institutt, Universitetet i Oslo P.O.Box 1053, Blindern, N-0316 Oslo, NORWAY

   E-mail address: ranestad@math.uio.no