Geometry of equivariant compactifications of 𝔾_{a}^{n}

Brendan Hassett and Yuri Tschinkel

June 29, 1999

1 Introduction

In this paper we begin a systematic study of equivariant compactifications of 𝔾_{a}^{n}. The question of classifying non-equivariant compactifications was raised by F. Hirzebruch ([10]) and has attracted considerable attention since (see [5], [15], [12] and the references therein). While there are classification results for surfaces and non-singular threefolds with small Picard groups, the general perception is that a complete classification is out of reach.

On the other hand, there is a rich theory of equivariant compactifications of reductive groups. The classification of normal equivariant compactifications of reductive groups is combinatorial. Essentially, the whole geometry of the compactification can be understood in terms of (colored) fans. In particular, these varieties do not admit moduli. For more details see [14], [4],

[2] and the references therein.

Our goal is to understand equivariant compactifications of 𝔾_{a}^{n}. The first step in our approach is to classify possible 𝔾_{a}^{n}-structures on simple varieties, like projective spaces or Hirzebruch surfaces. Then we realize general smooth 𝔾_{a}^{n}-varieties as appropriate (i.e., equivariant) blow-ups of simple varieties. This gives us a geometric description of the moduli space of equivariant compactifications of 𝔾_{a}^{n}.

In section 2 we discuss general properties of equivariant compactifications of 𝔾_{a}^{n} (𝔾_{a}^{n}-varieties). In section 3 we classify all possible 𝔾_{a}^{n}-structures on projective spaces ℙ^{n}. In section 4 we study curves, paying particular attention to non-normal examples. In section 5 we carry out our program completely for surfaces. In particular, we classify all possible 𝔾_{a}^{2}-structures on minimal rational surfaces. In section 6 we turn to threefolds. We give a classification of 𝔾_{a}^{3}-structures on smooth projective threefolds with Picard group of rank 1. Each section ends with a list of examples and open questions.

Acknowledgements. We are grateful to V. Batyrev, S. Bloch, T. de Fernex, V. Ginzburg, A. Iarrobino, D. Kazhdan and B. Totaro for useful discussions. The first author was partially supported by an NSF postdoctoral fellowship. The second author was partially supported by the NSA.

2 Generalities

We work in the category of algebraic varieties over F = \overline{ℚ}.

2.1 Definitions

Definition 2.1 Let 𝔾 be a connected linear algebraic group. An algebraic variety X admits a (left) 𝔾-action if there exists a morphism φ : 𝔾×X→X, satisfying the standard compatibility conditions. A 𝔾-variety X is a variety with a fixed (left) 𝔾-action such that the stabilizer of a generic point is trivial and the orbit of a generic point is dense.

For example, a normal 𝔾_{m}^{n}-variety is a toric variety.

Definition 2.2 A morphism of 𝔾-varieties is a morphism of algebraic varieties commuting with the 𝔾-action. A 𝔾-isomorphism is an isomorphism in the category of 𝔾-varieties. A 𝔾-equivalence is a diagram

𝔾 × X_{1} ⟶^{(α,j)} 𝔾 × X_{2}
↓  ↓
X_{1}  ⟶^{j} X_{2}

where α ∈ Aut(𝔾) and j is an isomorphism (in the category of algebraic varieties).

Clearly, every 𝔾-isomorphism is a 𝔾-equivalence. We shall omit 𝔾 if the group is understood.

One of our main observations in this paper is that classification of simple 𝔾-varieties up to equivalence, even projective spaces, is a non-trivial problem. This is in marked contrast to the situation for toric varieties. A toric variety admits a unique structure as a 𝔾_{m}^{n}-variety (up to equivalence). Here is a sketch of the argument for a projective toric variety X: Consider the connected component of the identity Aut(X)^{0} of the automorphism group Aut(X). This is an algebraic group which acts trivially on the Picard group of X. We pick a very ample line bundle and consider Aut(X)^{0} as a closed subgroup of the corresponding group PGL_{N}. In particular, Aut(X)^{0} is a linear algebraic group. The key ingredient now is the statement that all maximal tori in Aut(X)^{0} are conjugate (cf. 11.4 in [1]). Evidently, the maximal torus acts faithfully on X, and the action has a dense open orbit. This proves the claim.

Every 𝔾-variety contains an open subset isomorphic to 𝔾. We denote by D the complement of this open subset and call D the boundary. If X is normal, Hartog’s theorem implies that D must be a divisor (the complement to D is affine). Otherwise, we normalize and observe that the normalization is an isomorphism over 𝔾.

2.2 Line bundles and linearizations

Proposition 2.3 Let X be proper and normal algebraic variety. Then the action of 𝔾_{a}^{n} on the Picard group Pic(X) is trivial and every line bundle on X admits a unique linearization, up to scalar multiplication.

Proof. Uniqueness follows from Prop. 1.4 p. 33 in [13]. (The only relevant hypothesis is that X is geometrically reduced, which is part of our assumptions.) The proof of Prop. 1.5 p. 34 implies that every line bundle on X has a linearization. (Here we use that X is proper and that the Picard group of the group 𝔾 is trivial. To get the fact that the action of 𝔾 on the Pic(X) is trivial we need the normality of X.)

Corollary 2.4 Retain the assumptions of Proposition 2.3. Consider a basepoint free linear series W ⊂ H^{0}(X, L), stable under the action of 𝔾_{a}^{n}. Then the induced map f : X→ℙ(W^{∗}) (here we use the geometric convention) is 𝔾_{a}^{n}-equivariant.

Theorem 2.5 Let X be a proper and normal 𝔾_{a}^{n}-variety. Then Pic(X) is freely generated by the classes of the irreducible components D_{j} (j = 1, ..., t) of the boundary divisor D. The cone of effective Cartier divisors Λ_{eff}(X) ∈ Pic(X)_{ℝ} is given by

Λ_{eff}(X) = ⊕_{j=1}^{t}ℝ_{≥0}[D_{j}].

Proof. Choose an effective divisor A ⊂ X and consider the representation of 𝔾_{a}^{n} on the projectivization H^{0}(X, O(A)) (here we use that X is normal). This representation has a fixed point corresponding to an effective divisor supported at the boundary. To show that there are no relations between the classes [D_{1}], ..., [D_{t}] it suffices to observe that there exist no functions without zeros and poles on 𝔾_{a}^{n}.

Remark 2.6 Every effective cycle on a 𝔾-variety is rationally equivalent to a cycle supported on the boundary. (Here we are using the fact the 𝔾 is affine.)

2.3 Vector fields and the anticanonical line bundle

Theorem 2.7 Let X be a smooth and proper 𝔾-variety. Then the anticanonical class is a sum of classes of the irreducible components of the boundary D with coefficients which are all ≥ 1. If 𝔾 = 𝔾_{a}^{n} then the coefficients are all ≥ 2.

Proof. We first introduce some general terminology and exact sequences. Let Y be a smooth variety and B a smooth divisor. Let T_{Y} ⟨−B⟩ denote the sheaf whose sections are vector fields with logarithmic zeros along B. If x_{1}, …, x_{n} are local coordinates for Y so that B is given by x_{1} = 0, then local sections of T_{Y} ⟨−B⟩ take the form x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{n}} . There are two natural exact sequences

0 → T_{Y} ⟨−B⟩ → T_{Y} → N_{B/Y} → 0
0 → T_{Y} (−B) → T_{Y} ⟨−B⟩ → T_{B} → 0

where N_{B/Y} is the normal bundle to B.

Let v ∈ H^{0}(Y, T_{Y}) be a vector field. By definition, v vanishes normally to order one along B if its image in H^{0}(B, N_{B/Y}) is zero. If v arises from the action of a one-parameter group stabilizing B then it vanishes normally to order one along B. If v vanishes normally to order one along B, we can consider the corresonding element w ∈ H^{0}(B, T_{B}). If w = 0 then we say that v vanishes to order one. If v arises from a one-parameter group fixing B then it vanishes to order one. Generally, if v vanishes to order N − 1 (resp. normally to order N) then we can consider the corresponding element w ∈ H^{0}(B, N_{B/Y} (−(N − 1)B)) (resp. H^{0}(B, T_{B}(−(N − 1)B))). If w = 0 then we say that v vanishes normally (resp. vanishes) to order N along B; in particular, v may be regarded as a section of H^{0}(Y, T_{Y} ⟨−B⟩ (−(N − 1)B)) (resp. H^{0}(Y, T_{Y} (−NB))).

We now prove the theorem. First assume X is a 𝔾-variety and D_{i} an irreducible component of its boundary. We take Y ⊂ X as the complement to the singular locus of D_{i} and B = Y ∩ D_{i}. Let {v_{1}, …, v_{n}} ∈ H^{0}(X, T_{X}) be invariant vector fields spanning the Lie algebra of 𝔾, and M_{j} (resp. N_{j}) the order of vanishing (resp. normal vanishing) of v_{j} along B. By definition, N_{j} = M_{j} or M_{j} + 1. Consider the exterior power σ = v_{1} ∧ v_{2} ∧ … ∧ v_{n} ∈ H^{0}(X, Λ^{n}T_{X}); we bound (from below) the order of vanishing a_{i} of σ along D_{i}. Evidently a_{i} ≥ M_{1} + … + M_{n} but in fact slightly more is true. Using the adjunction isomorphism

Λ^{n}T_{Y} |B = Λ^{n−1}T_{B} ⊗ N_{B/Y}

we obtain

a_{i} ≥ min_{j=1,... ,n}(M_{1} + … + M_{j−1} + N_{j} + M_{j+1} + … + M_{n}).

For instance, since each N_{j} > 0 we obtain that a_{i} > 0. It follows that the canonical divisor is linearly equivalent to − ∑_{i} a_{i}D_{i}, where each a_{i} > 0.

Now assume that X is a 𝔾_{a}^{n}-variety. Let v_{1} be a vector field arising from the action of a one-parameter subgroup that fixes D_{i}, so that v_{1} vanishes to order one and M_{1} ≥ 1. We claim v_{1} necessarily vanishes normally to order two i.e. N_{1} ≥ 2. Consider the resulting element w ∈ H^{0}(B, N_{B/Y} (−B)) = End(N_{B/Y}), which exponentiates to give the induced 𝔾_{a}^{1} action on the normal bundle. Since 𝔾_{a}^{1} has no characters, this action is trivial and w = 0. The previous inequality guarantees that a_{i} ≥ 2. Hence the canonical divisor is linearly equivalent to − ∑_{i} a_{i}D_{i} where each a_{i} > 1.

Remark 2.8 The proof only uses the fact that X is smooth at generic points of the boundary divisor.

Corollary 2.9 Let X be a smooth projective 𝔾_{a}^{n}-variety with irreducible boundary D, which is fixed under the action. Then X = ℙ^{n}.

Proof. We see that M_{j} ≥ 1 for each j. It follows that K_{X} = −rD where r ≥ n + 1, i.e. X is Fano of index r ≥ n + 1. Hence X = ℙ^{n} and r = n + 1.

Corollary 2.10 Let X be a smooth projective 𝔾_{a}^{n}-variety with irreducible boundary D. Assume that the subgroup of 𝔾_{a}^{n} fixing D has dimension n − 1. Then X = ℙ^{n} or X = Q_{n}, the quadric hypersurface of dimension n.

Proof. After reordering, we obtain that M_{j} ≥ 1 for j = 1, …, n − 1. It follows from the proof of Theorem 2.7 that N_{j} ≥ 2 for j = 1, …, n − 1 and N_{n} ≥ 1. In particular, r ≥ n, i.e., X is Fano of index r ≥ n. Hence X = ℙ^{n} or Q_{n}.

2.4 A dictionary

Let L(𝔾_{a}^{n}) and U(𝔾_{a}^{n}) denote the Lie algebra and the enveloping algebra of 𝔾_{a}^{n}. Since 𝔾_{a}^{n} is commutative U(𝔾_{a}^{n}) is isomorphic to a polynomial ring in n variables. Let R = U(𝔾_{a}^{n})/I and assume that Spec(R) is supported at the origin; we use ℓ(R) and m_{R} to denote the length and the maximal ideal of R. Since I contains all the homogeneous polynomials of sufficiently large degree d, the elements of L(𝔾_{a}^{n}) act (via the regular representation) as nilpotent matrices on R. We can exponentiate to obtain an algebraic representation

ρ : 𝔾_{a}^{n} → Aut_{F} (R)

of dimension ℓ(R).

For concreteness, we introduce some additional notation. We consider L(𝔾_{a}^{n}) as a vector space over F with a distinguished basis S_{j} = \frac{∂}{∂x_{j}} so that U(𝔾_{a}^{n}) = F [S_{1}, …, S_{n}]. The representation ρ is obtained by multiplying by exp(x_{1}S_{1} + … + x_{n}S_{n}) ∈ R.

Proposition 2.11 Choose an F -basis {μ_{1}, μ_{2}, …, μ_{ℓ(R)}} for R. Then the coordinate functions f_{1}, f_{2}, …, f_{ℓ(R)} arising from the expansion

exp(x_{1}S_{1} + … + x_{n}S_{n}) = ∑_{j=1}^{ℓ(R)} f_{j}(x_{1}, …, x_{n})μ_{j}

form a basis for the solution space V of the system of partial differential equations

g[f(x_{1}, …, x_{n})] = 0 for each g ∈ I ⊂ F [\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}].

In particular, dim V = ℓ(R).

Proof. It suffices to exhibit a basis with the desired properties. This basis is constructed with Gröbner basis techniques [3] chapter 15. Consider homogeneous lexicographic order, which induces a total order on the set of all monomials in S_{1}, …, S_{n}. Let Init(I) be the initial ideal for I and {g_{i}} a Gröbner basis for I, i.e., the initial terms of the g_{i} generate Init(I). The monomials μ_{1}, μ_{2}, …, μ_{ℓ(R)} not contained in Init(I) form a basis for R (cf. proof of [3] Theorem 15.17.) Each monomial μ admits a unique representation in R

∑_{j=1}^{ℓ(R)} c_{j}(μ)μ_{j}.

The division algorithm (with respect to our Gröbner basis) implies that c_{j}(μ) = 0 whenever μ strictly precedes μ_{j} in the total order.

The formula

\frac{∂}{∂x_{i}} exp(x_{1}S_{1} + … + x_{n}S_{n}) = S_{i} exp(x_{1}S_{1} + … + x_{n}S_{n})

implies that the f_{j} are solutions to our system of partial differential equations.

We put a total order on the monomials in the x_{i}: x_{1}^{a(1)} … x_{n}^{a(n)} precedes x_{1}^{b(1)} … x_{n}^{b(n)} whenever S_{1}^{b(1)} … S_{n}^{b(n)} precedes S_{1}^{a(1)} … S_{n}^{a(n)}. We claim each f_{j} contains a term proportional to x_{1}^{m(1)} … x_{n}^{m(n)}, where μ_{j} = S_{1}^{m(1)} … S_{n}^{m(n)}, and this is the initial term of f_{j} with respect to our order. Since the f_{j} have distinct initial terms, they are linearly independent.

To prove the claim, note that exp(x_{1}S_{1} + … + x_{n}S_{n}) expands as a sum of nonzero terms

C_{a(1),... ,a(n)}x_{1}^{a(1)} … x_{n}^{a(n)}S_{1}^{a(1)} … S_{n}^{a(n)}.

Each term yields an element in R of the form

C_{a(1),... ,a(n)}x_{1}^{a(1)} … x_{n}^{a(n)} ∑_{j=1}^{ℓ(R)} c_{j}(S_{1}^{a(1)} … S_{n}^{a(n)})μ_{j},

where c_{j} = 0 whenever S_{1}^{a(1)} … S_{n}^{a(n)} precedes μ_{j}. In particular, the surviving nonzero terms

x_{1}^{a(1)} … x_{n}^{a(n)}S_{1}^{m(1)} … S_{n}^{m(n)}

all have the property that x_{1}^{a(1)} … x_{n}^{a(n)} precedes x_{1}^{m(1)} … x_{n}^{m(n)} in the order. Furthermore, the unique term

C_{m(1),... ,m(n)}x_{1}^{m(1)} … x_{n}^{m(n)}S_{1}^{m(1)} … S_{n}^{m(n)}

corresponds to the initial term of f_{j}.

To complete the proof, we show that

V := {f(x_{1}, …, x_{n}) : g[f] = 0 for each g ∈ I}

has dimension ℓ(R). Since I contains all the polynomials of degree d, each solution is polynomial with total degree < d. There is a natural pairing between F [x_{1}, …, x_{n}] and F [\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}]

⟨g, f⟩ = g[f]|_{(0,... ,0)}.

This induces a perfect pairing between homogeneous polynomials and operators of a given degree. Note that V = {f : ⟨g, f⟩ = 0 for each g ∈ I} which implies that dim V = ℓ(R).

We collect some basic properties of ρ:

Proposition 2.12 The 𝔾_{a}^{n}-representations ρ and V are dual, and ρ has a nondegenerate orbit (i.e., a cyclic vector) in R. The representation ρ is faithful iff Spec(R) ⊂ Spec(F [S_{1}, …, S_{n}]) is nondegenerate.

Proof. The vector space V has a natural 𝔾_{a}^{n}-action by translations. The first statement is clear from the preceding discussion. Consider the orbit of 1 ∈ R

ρ(x_{1}, …, x_{n}) · 1 = ∑_{i=1}^{ℓ(R)} f_{j}μ_{j}.

The linear independence of the f_{j} implies that this orbit is nondegenerate in R. The final statement is clear; indeed, S_{i} acts trivially iff Spec(R) ⊂ {S_{i} = 0}.

Remark 2.13 Note that V = ρ^{∗} has a natural structure as an R-module of length ℓ(R). Indeed, it coincides with the dualizing module ω_{R}. This can be seen using Macaulay’s method of inverse systems (see [3] chapter 21.2). It follows that ρ is self-dual iff R is Gorenstein.

A translation invariant subspace V ⊂ F [x_{1}, …, x_{n}] of dimension ℓ corresponds to a representation ρ : 𝔾_{a}^{n}→GL_{ℓ} with a fixed cyclic vector v. Indeed, we may regard the elements of V as coordinate functions on the nondegenerate orbit ρ(𝔾_{a}^{n}) · v. Consider the ideal I of constant coefficient differential operators annihilating V and write

R = F [\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}]/I.

Since I contains all monomials of sufficiently large degree, Spec(R) is supported at the origin. The pairing introduced in the proof of Proposition 2.11 may be used to show that Spec(R) has length ℓ.

We summarize our results in the following dictionary:

Theorem 2.14 There is a one-to-one correspondence among the following:

1. subschemes Spec(R) ⊂ Spec(F [\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}]) supported at the origin of length ℓ = ℓ(R);

2. translation invariant subspaces V ⊂ F [x_{1}, …, x_{n}] of dimension ℓ;

3. isomorphism classes of pairs (ρ, v) such that ρ : 𝔾_{a}^{n} → GL_{ℓ} is a representation and v is a cyclic vector (i.e., ρ(𝔾_{a}^{n}) · v is nondegenerate).

We now turn to a case of particular interest. Assume that S_{1}, …, S_{n} form a basis for the maximal ideal m_{R}, so that ℓ(R) = n + 1. Then the corresponding representation ρ_{R} is faithful and the induced action on ℙ(R) has a dense orbit. Hence for any Artinian local F -algebra, exponentiating the action of m_{R} on R yields a 𝔾_{a}^{ℓ(R)−1}-structure on ℙ(R) = ℙ^{ℓ(R)−1}. Conversely, assume we are given a 𝔾_{a}^{ℓ−1}-structure on ℙ^{ℓ−1}. By Proposition 2.3, this action admits a unique linearization on the line bundle O(+1), and we obtain a faithful representation ρ : 𝔾_{a}^{ℓ−1}→GL_{ℓ}. The corresponding ring of differential operators R is Artinian local of length ℓ. Since ρ is faithful, S_{1}, …, S_{ℓ−1} form a basis for m_{R}.

We summarize this discussion in the proposition:

Proposition 2.15 The following are equivalent:

1. Artinian local F -algebras R of length ℓ = ℓ(R), up to isomorphism;

2. equivalence classes of 𝔾_{a}^{ℓ−1}-structures on ℙ^{ℓ−1}.

2.5 Examples and questions

1. Not every point in the boundary D is contained in the closure of a 1-parameter subgroup. Construction: Blow up ℙ^{2} in a point at the boundary. Blow up again a point in the exceptional divisor. Every 1-parameter subgroup in ℙ^{2} is a line.

2. The theorem 2.7 fails for non-equivariant compactifications of 𝔾_{a}^{n}. For example, let X ⊂ ℙ^{2} × ℙ^{2} be a hypersurface of bidegree (1, d) with d ≥ 4. Then the anticanonical class O(2, 3 − d) is not contained in the interior of the effective cone.

3. Suppose X is a smooth projective 𝔾_{a}^{n}-variety with finitely many 𝔾_{a}^{n}- orbits. Is X rigid as an algebraic variety?

3 Projective spaces

In this section we study 𝔾_{a}^{n}-structures on projective spaces. Notice that every ℙ^{n} has a distinguished structure as a 𝔾_{a}^{n}-variety. The translation action on the affine space 𝔸^{n} extends to an action on ℙ^{n}, fixing the hyperplane at infinity. We denote this action by τ_{n}. It corresponds to the Artinian ring F [S_{1}, ..., S_{n}]/[S_{j}S_{j}, i, j = 1, ..., n]. It is easy to see that every 𝔾_{a}^{n}-structure on ℙ^{n} admits a specialization to τ_{n}.

In the following propositions, we classify 𝔾_{a}^{n}-structures on projective spaces of small dimension, up to equivalence (cf. 2.2). The first natural invariant is the Hilbert-Samuel function of the corresponding Artinian ring R, defined by χ_{R}(k) = ℓ(m_{R}^{k}/m_{R}^{k+1}).

Proposition 3.1 There is a unique 𝔾_{1}^{a}-structure on ℙ^{1}.

Proof. It is a consequence of Proposition 2.15.

Proposition 3.2 There are two distinct 𝔾_{a}^{2}-structures on ℙ^{2}. They are given by the following representations of 𝔾_{a}^{2}:

τ_{2}(a_{1}, a_{2}) = (\begin{matrix}1 & 0 & a_{2} \\ 0 & 1 & a_{1} \\ 0 & 0 & 1 \end{matrix})

and

ρ(a_{1}, a_{2}) = (\begin{matrix}1 & a_{1} & a_{2} + \frac{1}{2}a_{1}^{2} \\ 0 & 1 & a_{1} \\ 0 & 0 & 1 \end{matrix}) .

They correspond to the quotients of F [S_{1}, S_{2}] by the ideals I_{1} = [S_{1}S_{2}, S_{2}^{2}, S_{1}^{2}] and I_{2} = [S_{1}S_{2}, S_{2} − S_{1}^{2}].

Proof. It suffices to classify Artinian local F -algebras R of length three up to isomorphism. If the tangent space has dimension two then R = F [S_{1}, S_{2}]/I_{1}. If the tangent space to R has dimension one then R = F [S_{1}, S_{2}]/I_{2} (it is clear that this is the only one). These correspond to the representations τ_{2} and ρ respectively.

Proposition 3.3 There are four distinct 𝔾_{a}^{3}-structures on ℙ^{3}. They correspond to the quotients of F [S_{1}, S_{2}, S_{3}] by the following ideals:

I_{1} = [S_{1}^{2} − S_{2}, S_{1}S_{2} − S_{3}, S_{1}S_{3}]
I_{2} = [S_{1}^{2} − S_{2}, S_{1}S_{2}, S_{1}S_{3}]
I_{3} = [S_{1}^{2}, S_{1}S_{2} − S_{3}, S_{2}^{2}]
I_{4} = [S_{1}^{2}, S_{1}S_{2}, S_{2}^{2}, S_{2}S_{3}, S_{3}^{2}, S_{1}S_{3}].

Proof. If the tangent space to R has dimension one or three, then R is necessarily the quotient of the polynomial ring by I_{1} or I_{3}. If the tangent space has dimension two then ℓ(m_{R}^{2}/m_{R}^{3}) = 1. Consider the nonzero symmetric quadratic form

q : m_{R}/m_{R}^{2} × m_{R}/m_{R}^{2}→m_{R}^{2}/m_{R}^{3}.

Let S_{3} generate m_{R}^{2}. Then there exist S_{1}, S_{2} ∈ m_{R} spanning m_{R}/m_{R}^{2} such that the quadratic form equals S_{1}S_{2} or S_{1}^{2}.

Proposition 3.4 There are ten distinct 𝔾_{a}^{4}-structures on ℙ^{4}. They correspond to the quotients of F [S_{1}, S_{2}, S_{3}, S_{4}] by the following ideals:

I_{1} = [S_{1}^{2} − S_{2}, S_{1}S_{2} − S_{3}, S_{1}S_{3} − S_{4}, S_{2}S_{3}, S_{1}S_{4}]
I_{2} = [S_{1}^{2} − S_{3}, S_{1}S_{2}, S_{2}^{2}, S_{1}S_{3} − S_{4}, S_{1}S_{4}]
I_{3} = [S_{1}^{2} − S_{3}, S_{1}S_{2}, S_{2}^{2} − S_{3}, S_{1}S_{3} − S_{4}, S_{1}S_{4}]
I_{4} = [S_{1}^{2} − S_{3}, S_{1}S_{2} − S_{3}, S_{2}^{2}, S_{1}S_{3} − S_{4}, S_{1}S_{4}]
I_{5} = [S_{1}^{2} − S_{3}, S_{1}S_{2} − S_{4}, S_{2}^{2}, S_{1}S_{3}, S_{1}S_{4}, S_{2}S_{3}]
I_{6} = [S_{1}^{2} − S_{3}, S_{2}^{2} − S_{4}, S_{1}S_{2}, S_{1}S_{3}, S_{2}S_{4}]
I_{7} = [S_{1}^{2} − S_{4}, S_{2}S_{3} − S_{4}, S_{1}S_{2}, S_{1}S_{3}, S_{2}^{2}, S_{3}^{2}, S_{1}S_{4}]
I_{8} = [S_{1}^{2}, S_{2}^{2}, S_{3}^{2}, S_{1}S_{2} − S_{4}, S_{1}S_{3}, S_{2}S_{3}]
I_{9} = [S_{1}^{2} − S_{4}, S_{2}^{2}, S_{3}^{2}, S_{1}S_{2}, S_{1}S_{3}, S_{1}S_{4}, S_{2}S_{3}]
I_{10} = [S_{i}S_{j}, i, j = 1, ..., 4]

Proof. We consider the possible shapes of the Hilbert-Samuel function χ_{R}. In the cases where χ_{R}(1) = 1 or 4 it is clear that the only possibilities are I_{1} and I_{10} respectively.

Assume that χ_{R}(1) = 2, χ_{R}(2) = 1 and χ_{R}(3) = 1. Choose S_{4} ∈ m_{R}^{3}, S_{3} ∈ m_{R}^{2}, and S_{1}, S_{2} ∈ m_{R} which span the maximal ideal of the graded ring associated to R. For a suitable choice of S_{2} and S_{4}, we may assume that S_{2}S_{3} = 0 and S_{1}S_{3} = S_{4}. Consider the map s_{2} : m_{R}/m_{R}^{2}→m_{R}^{2}/m_{R}^{3} induced by multiplying by S_{2}. If s_{2} = 0 then we may choose S_{3} so that S_{1}^{2} = S_{3}; we obtain I_{2}. If s_{2} ≠ 0 and S_{2} ∉ ker(s_{2}) then we may choose S_{1} so that S_{1}S_{2} = 0. However, S_{4} ≠ 0 implies S_{1}^{2} ≠ 0, so after rescaling S_{1}, S_{2}, and S_{4} we obtain I_{3}. If s_{2} ≠ 0 and S_{2} ∈ ker(s_{2}) then (after rescaling S_{3}) we obtain S_{1}S_{2} = S_{3}. Again, after rescaling S_{1} and S_{2}, S_{1}^{2} ≠ 0 and we obtain I_{4}.

Assume that χ_{R}(1) = 2 and χ_{R}(2) = 2. Then we choose S_{1}, ..., S_{4} such that S_{3} and S_{4} are in m_{R}^{2} and S_{1}, S_{2} are independent modulo m_{R}^{2}. The ring structure on R is determined by the vector-valued quadratic form

q : m_{R}/m_{R}^{2} × m_{R}/m_{R}^{2}→m_{R}^{2}/m_{R}^{3}.

This corresponds to choosing a codimension 1 subspace of Sym^{2}(m_{R}/m_{R}^{2}). Up to changes of coordinates in S_{1} and S_{2}, each such subspace is spanned by vectors {S_{1}^{2}, S_{2}^{2}} and {S_{1}S_{2}, S_{1}^{2}}. This gives the cases I_{5} and I_{6}.

Assume that χ_{R}(1) = 3 and χ_{R}(2) = 1. This corresponds to I_{7}, I_{8}, and I_{9}. The ring structure is determined by the quadratic form q (with values in F), which has rank 3, 2 or 1.

Proposition 3.5 There are finitely many distinct 𝔾_{a}^{5}-structures on ℙ^{5}.

Proof. These are written out explicitly in Suprunenko [16] pp. 136-150.

The above discussion mirrors the classification of algebras of commutative nilpotent matrices in the book [16]. (Notice a misprint in the classification of algebras corresponding to 𝔾_{a}^{3}-structures on ℙ^{3} on page 134). The arguments in this book yield a classification of Artinian algebras of length n + 1 with the following Hilbert-Samuel functions (though the author does not make this explicit):

\begin{matrix}χ_{R}(0) & χ_{R}(1) & χ_{R}(2) & χ_{R}(3) & ... & χ_{R}(n) \\ 1 & n & 0 & 0 & ... & 0 \\ 1 & 1 & 1 & 1 & ... & 1 \\ 1 & 2 & 1 & 1 & ... & 0 \\ 1 & 3 & 1 & 1 & ... & 0 \\ 1 & 2 & 2 & 1 & ... & 0 \end{matrix}

In particular, there are finitely many algebras with each of these Hilbert-Samuel functions χ_{R}. This suffices to obtain a complete classification of 𝔾_{a}^{n} structures on ℙ^{n} for n ≤ 5.

Beginning with dimension 6 we obtain moduli. As an example, let us consider Artinian rings with Hilbert-Samuel function of the shape (1, n − k, k, 0, ..., 0) (for suitable k). These correspond to k-dimensional spaces of quadratic forms in n − k variables (up to coordinate transformations of the n − k variables). The quadratic forms are obtained by dualizing the natural map

Sym^{2}(m_{R}/m_{R}^{2})→m_{R}^{2}/m_{R}^{3}.

For example, if n = 6 and k = 2 the moduli space is birational to the moduli space of elliptic curves.

Example 3.6 There exists at least one 1-parameter family of inequivalent 𝔾_{a}^{6}-structures on ℙ^{6}.

If n = 8 and k = 3 we get the moduli space of genus 5 curves. If n = 9 and k = 3 we obtain the moduli space of K3 surfaces of degree 8. The appearance of these K3 surfaces and the genus 5 curves is quite interesting. Can it be explained geometrically, in terms of birational maps between different 𝔾_{a}^{n}- structures?

For a good general introduction to Artinian rings (and many further references) see [11].

Proposition 3.7 The projective space ℙ^{n} admits a unique 𝔾_{a}^{n}-structure with finitely many orbits. It corresponds to the Artinian ring F [S_{1}, ..., S_{n}]/I, where

I = [S_{1}^{2} − S_{2}, S_{1}S_{2} − S_{3}, ..., S_{1}S_{n−1} − S_{n}, S_{i}S_{j}, i + j > n]
≃ [S_{1}^{i} − S_{i}, S_{i}S_{j}, i + j > n].

Proof. There is a unique fixed point under the 𝔾_{a}^{n}-action. Projecting from it gives a ℙ^{n−1} with finitely many 𝔾_{a}^{n−1}-orbits. By the inductive hypothesis, this ℙ^{n−1} has the indicated structure. The Artinian ring R(ℙ^{n−1}) for ℙ^{n−1} is a quotient of the Artinian ring R(ℙ^{n}). Let S_{n} ∈ R(ℙ^{n}) be a non-zero element mapped to 0 ∈ R(ℙ^{n−1}). Since R(ℙ^{n−1}) ≃ F [T_{1}]/[T_{1}^{n}], there exists an element S_{1} ∈ R(ℙ^{n−1}) such that S_{1}^{n−1} ≠ 0. Then S_{1}^{n} = cS_{n} for some constant c ∈ F. If c = 0 the action has infinitely many orbits. Otherwise, (after rescaling) we obtain the desired ideal.

3.1 Examples and questions

1. Untwisting different actions on ℙ^{2}: Take ℙ^{2} with the ρ-action and choose a generic one-parameter subgroup. Let C be the conic obtained as the closure of a generic orbit of the one-parameter subgroup. The curve C is tangent to the line at infinity at the fixed point p. Blow up the fixed point on C 3 times. Contract the strict transforms of the line at infinity, and the first two exceptional curves.

2. In general, there does not exist an irreducible variety parametrizing all 𝔾_{a}^{n}-structures on ℙ^{n} (fails for n = 7). This is related to the fact that a general length 8 subscheme of 𝔸^{4} is not a limit of 8 distinct points (this was pointed out to us by Iarrobino, see [11]). For example, consider the subschemes cut out by 7 general quadrics in 4 homogeneous variables. These subschemes deform only to subschemes of the same type. Clearly, not every Artinian local F -algebra of length 8 has tangent space of dimension ≥ 4 (the curvilinear ones have 1-dimensional tangent space).

3. Give an explicit factorization for 𝔾_{a}^{n}-equivariant birational automorphisms of the projective space ℙ^{n}.

4. Give a dictionary between 𝔾_{a}^{n}-structures on smooth quadrics Q_{n} and certain Artinian rings (with additional structure).

4 Curves

Proposition 4.1 Every smooth proper 𝔾_{a}^{1}-variety is isomorphic to ℙ^{1} with the standard translation action τ_{1}.

Proof. Exercise.

Lemma 4.2 Let V be the standard representation of 𝔾_{a}^{1}. Then Sym^{n}(V) has a filtration

0 ⊂ F_{0} ⊂ F_{1} ⊂ ... ⊂ F_{n} = Sym^{n}(V)

which is compatible with the 𝔾_{a}^{1}-action and such that F_{i} ≃ Sym^{i}(V) and F_{i+1}/F_{i} is the 1-dimensional trivial representation. Furthermore, every stable subspace of Sym^{n}(V) arises in this way.

Proof. Exercise.

Remark 4.3 The Jordan canonical form gives us a complete description of representations of 𝔾_{a}^{1}. They are isomorphic to direct sums of the representations Sym^{n}(V).

Proposition 4.4 Every proper 𝔾_{a}^{1}-variety C with an equivariant projective embedding is isomorphic to ℙ^{1} embedded by a complete linear series with the translation action.

Proof. Clearly, the normalization of the curve C is isomorphic to ℙ^{1}. Furthermore, the normalization map ν : ℙ^{1}→C is equivariant and an isomorphism away from the fixed point P_{∞} ∈ ℙ^{1}. The morphism ν is given by some basepoint free linear series W on ℙ^{1}, which is stable under the 𝔾_{a}^{1}-action. In particular, W ⊂ H^{0}(O_{ℙ^{1}}(n)) ≃ Sym^{n}(V) (where V is the standard representation). By the previous Lemma, each stable proper subspace of Sym^{n}(V) corresponds to a linear series on ℙ^{1} with basepoints. This concludes the proof.

Our next goal is to classify proper 1-dimensional 𝔾_{a}^{1}-varieties C. Clearly, the normalization C̃ has to be isomorphic to ℙ^{1} with the standard action φ and with the conductor-ideal vanishing at the fixed point t = 0. Hence it suffices to classify conductor-ideals I ⊂ F [t]|_{t=0}, stable under the group action.

Theorem 4.5 The only conductor-ideals I which are stable under the group action φ are preimages of some semigroup Σ ⊂ ℤ_{<0} under the valuation homomorphism.

Proof. Consider the complete local ring F [[t]] with maximal ideal m. We make the identification

\frac{m}{m^{n+2}} ≃ Sym^{n}(V)

(it follows from the definition of the action φ : t ↦ t · (1 − at + a^{2}t^{2} − ...) extended to the completion). By lemma 4.2, all subspaces stable under the action of 𝔾_{a} coincide with preimages of subsets of the valuation group.

4.1 Examples and questions

1. Let C ∈ ℙ^{2} be a cuspidal cubic plane curve. Then it is a proper 𝔾_{a}^{1}- variety which does not admit 𝔾_{a}^{1}-equivariant projective embeddings. Indeed, the action on the normalization ℙ^{1} = C̃ of C is given by

φ : t ↦ \frac{t}{1 + at}.

Note that the underlying topological space of C is just ℙ^{1} and that all local rings coincide, except at the cusp 0. The local ring O_{C,0} is equal to the ideal generated by t^{2}, t^{3} in the ring F [t]|_{t=0}. This ideal is fixed under the action of φ. Therefore, the action descends to C. By (4.4), C does not admit a 𝔾_{a}^{1}-equivariant projective embedding.

2. Describe versal deformation spaces of non-normal proper 𝔾_{a}^{1}-varieties together with the 𝔾_{a}^{1}-action on these spaces.

5 Surfaces

Throughout this section X will be a smooth proper 𝔾_{a}^{2}-variety.

Proposition 5.1 Let E ⊂ X be a (−1)-curve. Then there exists a morphism of 𝔾_{a}^{2}-varieties X→X′ which blows down E.

Proof. This follows from proposition 2.3 and corollary 2.4.

Proposition 5.2 Every 𝔾_{a}^{2}-surface X admits a 𝔾_{a}^{2}-equivariant morphism onto ℙ^{2} or a Hirzebruch surface 𝔽_{n}.

Proof. This follows from the existence of minimal models for rational surfaces.

5.1 Hirzebruch surfaces

Let X be a 𝔾_{a}^{2}-variety. Assume that X is isomorphic to 𝔽_{n} as an algebraic variety with n > 0. Its zero-section e is stabilized under the group action, as is the distinguished fiber f. Let ξ_{n} be the line bundle on X corresponding to the section at infinity. There is an induced equivariant morphism

μ : X → ℙ(H^{0}(X, ξ_{n})^{∗}) = ℙ^{n+1}.

The image μ(X) is the cone over a smooth rational normal curve of degree n; μ contracts e to the vertex of this cone.

We compute the representation of 𝔾_{a}^{2} on H^{0}(X, ξ_{n})^{∗}. It has a distinguished one-dimensional fixed subspace W_{1} corresponding to the vertex. The resulting representation on H^{0}(X, ξ_{n})^{∗}/W_{1} can be easily understood geometrically. It has a one dimensional kernel and the corresponding faithful representation

𝔾_{a}^{1} → GL(H^{0}(X, ξ_{n})^{∗}/W_{1})

is the n-fold symmetric power of the standard two-dimensional representation. Here we are using the fact that μ(X) is the cone over a rational normal curve of degree n.

Choose a basis S_{1}, S_{2} ∈ L(𝔾_{a}^{2}) such that S_{1} acts nontrivially and S_{2} acts trivially as matrices on H^{0}(X, ξ_{n})^{∗}/W_{1}. As a matrix on H^{0}(X, ξ_{n})^{∗}, S_{2} has image contained in W_{1}, and S_{1}S_{2} = 0. We have already seen that S_{1}^{n} ≠ 0 as a matrix on H^{0}(X, ξ_{n})^{∗}/W_{1}.

We consider two possible cases: either S_{1}^{n+1} ≠ 0 or S_{1}^{n+1} = 0. In the first case, we apply the following fact about nilpotent matrices.

Lemma 5.3 Let S_{1} be an (n+2)×(n+2) nilpotent matrix such that S_{1}^{n+1} ≠ 0. Then the centralizer of S_{1} consists of the algebra of matrices generated by S_{1} and the identity.

Proof. This follows from a straightforward induction once we put S_{1} in Jordan canonical form.

The lemma implies that S_{2} may be written as some polynomial of S_{1}. The fact that the images of S_{2} and S_{1}^{n+1} both lie in W_{1} implies that T = cS^{n+1} for some c ≠ 0. In this case we have

H^{0}(X, ξ_{n})^{∗} = ρ_{R} where R = F [S_{1}, S_{2}]/[S_{1}S_{2}, S_{2} − S_{1}^{n+1}].

Furthermore, S_{2} acts nontrivially on the distinguished fiber f.

We now assume that S_{1}^{n+1} = 0. In this case we have

H^{0}(X, ξ_{n})^{∗} = ρ_{R} where R = F [S_{1}, S_{2}]/[S_{1}S_{2}, S_{1}^{n+1}],

and the action is trivial along the distinguished fiber.

In conclusion:

Proposition 5.4 Let X be a 𝔾_{a}^{2}-variety as above and let ξ_{n} denote the line bundle corresponding to the section at infinity. If the action on the distinguished fiber is nontrivial then the representation

𝔾_{a}^{2} → GL(H^{0}(X, ξ_{n})^{∗}) = GL_{n+2}

is equivalent to exp(a_{1}S_{1} + a_{n+1}S_{2}) where S_{2} = S_{1}^{n+1} ≠ 0 and S_{1}S_{2} = 0. If the distinguished fiber is fixed under the action then the representation is equivalent to exp(a_{1}S_{1} + a_{n+1}S_{2}) where S_{1}^{n+1} = 0 and S_{1}S_{2} = 0.

A geometrical interpretation is obtained as follows. Let W be the (n+2)- dimensional representation of 𝔾_{a}^{2} described above. Then each X admits an equivariant birational morphism into ℙ(W) and corresponds to the closure of some nondegenerate orbit. To classify the surfaces X it suffices to classify the nondegenerate 𝔾_{a}^{2} orbits of ℙ(W) modulo automorphisms, i.e. the 𝔾_{a}^{2}- automorphisms of ℙ(W).

In the first case, these are exactly the automorphisms commuting with the action of S_{1}, i.e., the homotheties and the matrices

exp(a_{1}S_{1} + a_{2}S_{1}^{2} + … + a_{n+1}S_{1}^{n+1}).

Note that this gives ℙ(W) the structure of a 𝔾_{a}^{n+1}-variety, which has a dense open orbit equal to the complement of the distinguished hyperplane. In particular, any two nondegenerate orbit closures in ℙ(W) are related by an automorphism of ℙ(W). It follows that X is unique up to equivalence.

In the second case, these automorphisms include the homotheties and the matrices

exp(a_{1}S_{1} + a_{2}S_{1}^{2} + … + a_{n}S_{1}^{n} + a_{n+1}S_{2}).

Again, ℙ(W) has the structure of a 𝔾_{a}^{n+1}-variety, and any two nondegenerate orbit closures are related by an automorphism. It follows that X is unique up to equivalence.

These arguments yield the following:

Proposition 5.5 The Hirzebruch surfaces 𝔽_{n} with n > 0 each have two distinct 𝔾_{a}^{2}-structures, including a unique structure with a nontrivial action on the distinguished fiber. The second structure is obtained by taking an elementary transformation of the structure on 𝔽_{n−1}. The product 𝔽_{0} = ℙ^{1}×ℙ^{1} has a unique 𝔾_{a}^{2}-structure, induced from the 𝔾_{a}^{1} actions on each factor.

The elementary transformation involves blowing-up the intersection of the zero section and the distinguished fiber of 𝔽_{n−1}, and then blowing-down the proper transform of this fiber. To prove the last statement, we project from a fixed point of 𝔽_{0}. The image is ℙ^{2} with 𝔾_{a}^{2} acting by translation.

5.2 Examples and questions

1. Interesting singular surfaces admitting 𝔾_{a}^{2}-structures: Del Pezzo surface of degree 5 with an isolated A_{4}-singularity.

2. Can the 𝔾_{a}^{2}-structures on a given (smooth) surface have moduli?

3. Classify 𝔾_{a}^{2}-structures on projective surfaces with log-terminal singularities and Picard number 1.

6 Threefolds

Theorem 6.1 Let X be a smooth projective 𝔾_{a}^{3}-variety such that the boundary D is irreducible. Then X is one of the following:

1. X = ℙ^{3}, D a hyperplane (the possible 𝔾_{a}^{3}-structures were listed in 3.3);

2. Q_{3} ⊂ ℙ^{4} is a smooth quadric, D a tangent hyperplane section. It has a unique 𝔾_{a}^{3}-structure (described in the proof).

Proof. We know that −K_{X} = r · D where r ≥ 2 (2.7). Therefore, X is a Fano variety of index r ≥ 2 and it is rational. Furthermore, if it has index = 2 then the subgroup fixing D has dimension one (by Corollary2.10). We first consider the case where the index > 2.

We show there is a unique 𝔾_{a}^{3}-structure on Q_{3}, and that the boundary D is necessarily equal to a tangent hyperplane section. First let us convince ourselves that a quadric with a tangent hyperplane is equivariant. Consider ℙ^{3} with the translation action τ_{3}, which fixes the hyperplane at infinity P.

Blow up a smooth conic curve C ⊂ P ⊂ ℙ^{3} and blow down the proper transform of P. Now we prove that there are no other 𝔾_{a}^{3}-structures on Q_{3}. Any 𝔾_{a}^{3}-action on Q_{3} has a fixed point p. Projecting from p gives an equivariant birational map f : Q_{3} −−→ ℙ^{3}. The induced map from the blowup of Q_{3} at p to ℙ^{3} is the blow-up of a conic contained in the proper transform of D (where D is the boundary of 𝔾_{a}^{3} in Q_{3}). The proper transform of D is a plane P. The classification of 𝔾_{a}^{3}-structures on ℙ^{3} implies that P is fixed under the action of 𝔾_{a}^{3} on ℙ^{3} (cf. 3.3).

We return to the case where index equals two. By Furushima’s classification of non-equivariant compactifications of 𝔾_{a}^{3} (cf. [5], [6]), X is a codimension 3 linear section of the Grassmannian Gr(2, 5). Consider the action induced on F, the variety of lines on X. Any line on X has normal bundle equal to O ⊕ O or O(−1) ⊕ O(+1) and there is always a line of the second type (cf. [8]). Choose such a line L stable under the 𝔾_{a}^{3} action (we are choosing a fixed point on the locus of lines of the second type in F). Projecting from L gives an equivariant birational map

π_{L} : X −−→ Q_{3}

(cf. pp. 112 of [8]).

By [6], there are two cases to consider. In the first case the boundary D ⊂ X is non-normal, with singular locus L. The total transform of D consists of a hyperplane section H ⊂ Q_{3}. The image of the boundary D is a smooth rational curve of degree 3, contained in H. We have already shown that there is a unique 𝔾_{a}^{3}-structure on Q_{3}, which does not admit any smooth rational curves of degree 3 contained in the boundary and stable under the action (the only stable curve in H is the distinguished ruling).

In the second case the boundary D ⊂ X is a normal singular Del Pezzo surface of degree 5 with an isolated A_{4}-singularity. The curve L ⊂ D is the unique (−1)-curve in the minimal resolution of D. Under π_{L}, D is mapped birationally (and equivariantly!) to a tangent hyperplane section H ⊂ Q_{3}. The subgroup fixing H has dimension 2, so the same holds for D. This contradicts Corollary 2.10.

6.1 Examples and questions

1. A singular projective 𝔾_{a}^{3}-variety with one irreducible boundary divisor on which the 𝔾_{a}^{3}-action is trivial. Construction: Blow up a pair of intersecting lines in ℙ^{3}. Then blow down the proper transform of the plane containing them. The resulting variety is a singular quadric hypersurface in ℙ^{4}.

2. 𝔾_{a}^{3}-equivariant flop: Consider a quadric hypersurface Q^{∗} ⊂ ℙ^{4} with an isolated singularity p. Let Y be the blow up of Q^{∗} at p. Let E ≡ ℙ^{1} × ℙ^{1} be the exceptional divisor. Blowing down E in different directions yields smooth threefolds X_{1} and X_{2}. It suffices to exhibit a 𝔾_{a}^{3}-structure on Q^{∗} (all the constructions are natural and equivariant). This structure is obtained by using the fact that Q^{∗} is a cone over 𝔽_{0}.

3. 𝔾_{a}^{3}-equivariant flip: Consider the cone V over the Veronese surface ℙ^{2} ⊂ ℙ^{5}. There exists a 𝔾_{a}^{3}-structure on V with a non-singular fixed point p. Indeed, given ℙ^{2} with the translation action the cone V has a unique 𝔾_{a}^{3}-structure with fixed boundary divisor. Let Y be the blowup of V with center in p and let Z be the variety obtained from Y by contracting the proper transform of the ruling through p. In particular, Z is obtained from Y as a small contraction and K_{Z} is not ℚ-Cartier. The variety Z is isomorphic to the cone over a cubic scroll 𝔽_{1} ⊂ ℙ^{4}. The flipped threefold X is a small resolution of Z.

4. If X is a smooth projective 𝔾_{a}^{3}-variety and the action on the boundary is trivial is X = ℙ^{3}?

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Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637 USA email address: hassett@math.uchicago.edu

Department of Mathematics U.I.C. 851 S. Morgan Str. Chicago, IL 60607 USA email address: yuri@math.uic.edu