ON CERTAIN LOCI OF CURVES OF GENUS g ≥ 4 WITH

WEIERSTRASS POINTS WHOSE FIRST NON–GAP IS THREE

G. Casnati and A. Del Centina

1. Introduction and notations

Let 𝔐_{g} be the moduli space of smooth curves of genus g ≥ 4 over the complex field ℂ and let 𝔗_{g} ⊆ 𝔐_{g} be the trigonal locus, i.e. the set of points [C] ∈ 𝔐_{g} representing trigonal curves C of genus g ≥ 4. Recall that each such curve C carries exactly one g_{3}^{1} (resp. at most two) if g ≥ 5 (resp. g = 4). Let |D| be a g_{3}^{1} on C and suppose that it has a total ramification point at P (t.r. for short), i.e. that there is on C a point P such that 3P ∈ |D|. Such a P is a Weierstrass point whose first non–gap is three. In the present paper we study some sub–loci of 𝔗_{g} related to curves possessing such points.

Let us be more precise. Following [Co1], we say that |D| is of kind n if h^{0}(C, O_{C}(nD)) = n + 1 but h^{0}(C, O_{C}((n + 1)D)) > n + 2. All the g_{3}^{1} on a curve C are of the same kind.

From [Co1], lemma 2.4 and corollary 2.11, it follows that

(1.1)  \frac{g − 1}{3} ≤ n ≤ \frac{g}{2},

and there are trigonal curves C with |D| of kind n for each n in this range.

It is known that 𝔗_{g} ⊆ 𝔐_{g} is an irreducible, unirational subvariety of dimension 2g + 1 (see for instance [B–DC]) and that it is actually rational for certain values of g (see [SB]).

Let

𝔗_{g,n}(α) := { [C] ∈ 𝔗_{g} | one of the g_{3}^{1} on C is of type n,

has at least α t.r. }.

We have that

𝔗_{g}(α) := ⋃_{n} 𝔗_{g,n}(α) ⊆ 𝔐_{g}

is the moduli space of trigonal curves of genus g having at least α t.r., or if g ≥ 5, under another viewpoint it is the moduli space of smooth curves having at least α Weierstrass points whose first non–gap is three.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 14H05, 14H45.

Key words and phrases. Curve, Moduli, Rationality.

The first author has been supported by the framework of the SCIENCE contract n. SCI–0398–C(A).

Both the author has been partially supported by MURST in the framework of the national project:

“Geometria Algebrica, Algebra Commutativa ed Aspetti Computazionali”.

It is known that 𝔗_{g}(1) is irreducible (see [Ar]) and unirational (see [Co3], theoreem 3.3). The authors do not know any analogous result for 𝔗_{g}(α) when α ≥ 2. Some results are proved in [B–S] where a stratification of 𝔗_{g,n} is defined and the irreducibility of some strata is proved. None of the strata defined in [B–S] coincide with 𝔗_{g,n}(α) (see also section 2).

In section 2 we prove the following

Theorem. If g ≥ 4 and α ≤ 2 then 𝔗_{g,n}(α) is irreducible and unirational.

Our main results are proved in sections 3 and 4 and can be summarized in the following statement.

Theorem. If g ≥ 4 and α = 1, 2, then 𝔗_{g,[g/2]}(α) is a rational variety.

In view of the irreducibility of 𝔗_{g}(1), the above result implies that 𝔗_{g}(1) is actually rational. Notice that these results give also partial answers to question 1 raised in [E–H].

We remark that it is plausible that ⋃_{n} 𝔗_{g,n}(α) ⊆ \overline{𝔗_{g,[g/2]}(α)} for α = 2 (this is true for α = 0, 1). In this case our results would imply the rationality of 𝔗_{g}(α) also for α = 2.

Finally we focus our attention on the case g = 4 proving the rationality of certain sub–loci of 𝔐_{4}. In particular we get, as a by–product, that

Theorem. The generators of the Chow ring A(𝔐_{4}) described in [Fa] are rational.

Notations. PGL_{n} is the general projective linear group on ℂ.

If g is an element of a certain group (resp. vector space) G then ⟨g⟩ denotes the subgroup (resp. subspace) of G generated by g.

We denote by ≅ isomorphisms and by ≈ birational equivalences.

For all the other notations and definition we always refer to [Ha].

2. The loci 𝔗_{g,n}(α)

We begin by recalling some easy and helpful facts about the curves represented by points of 𝔗_{g,n}(α). Take complementary subspaces ℙ_{ℂ}^{n−1}, ℙ_{ℂ}^{g−n−1} ⊆ ℙ_{ℂ}^{g−1} and let R_{n−1} ⊆ ℙ_{ℂ}^{n−1}, R_{g−n−1} ⊆ ℙ_{ℂ}^{g−n−1} be rational normal curves. Fix an isomorphism φ: R_{n−1} → R_{g−n−1} and consider

S_{n−1,g−n−1} := ⋃_{x∈R_{n−1}}⟨x, φ(x)⟩.

S_{n−1,g−n−1} is a ruled surface called a rational normal scroll. In any case the minimal resolution of singularities of S_{n−1,g−n−1} is the Hirzebruch surface

𝔽_{g−2n} := ℙ(O_{ℙ_{ℂ}^{1}} ⊕ O_{ℙ_{ℂ}^{1}}(2n − g)).

The surface S_{n−1,g−n−1} is singular if and only if n = 1. If n ≥ 2 then

Pic(S_{n−1,g−n−1}) = ℤH ⊕ ℤf

where H is any hyperplane section and f is a line. We have R_{n−1} ∼ H − (g − n − 1)f, R_{g−n−1} ∼ H − (n − 1)f.

ON CERTAIN LOCI OF CURVES OF GENUS g ≥ 4 3

If [C] ∈ 𝔗_{g,n}, then the canonical model of C lies on a unique rational normal scroll S_{n−1,g−n−1}, which rulings cut out on C the g_{3}^{1}.

One has C ∈ |3H + (4 − g)f| and dim |3H + (4 − g)f| = 2g + 7. Let U ⊆ |3H + (4 − g)f| be the open subset of smooth curves. Then U ≠ ∅ and each element of U represents a point in 𝔗_{g,n} (see for instance [B–DC]).

Now fix g and n satisfying (1.1). First we assume that g ≥ 5, hence n ≥ 2 and the surface S := S_{n−1,g−n−1} ⊆ ℙ_{ℂ}^{g−1} is smooth. Fix α pairwise distinct fibres f_{1}, …, f_{α} of S. For each (P_{1}, …, P_{α}) ∈ X := f_{1} × ⋯ × f_{α} consider the projective subspace Σ_{(P_{1},...,P_{α})} ⊆ |3H + (4 − g)f| generated by

{ C ∈ U | C · f_{i} = 3P_{i}, i = 1, …, α }.

Then we have the scheme Σ(X) := ⋃_{(P_{1},...,P_{α})∈X} Σ_{(P_{1},...,P_{α})} and its natural projection p: Σ(X) → f_{1} × ⋯ × f_{α}.

Lemma 2.1. Let g ≥ 5. If α ≤ 2 then Σ(X) is irreducible.

Proof. In view of theorem 8 of section 6, chapter I of [Sha] it suffices to check that Σ_{(P_{1},...,P_{α})} has constant dimension. More precisely we have dim(Σ_{P}) = 2g + 4 and dim(Σ_{(P_{1},P_{2})}) = 2g + 1.

If n = g/2, and C ∈ Σ_{(P_{1},P_{2})} then C is a curve of bidegree (3, n+1) on a smooth quadric Q, with two flexes at the points P_{1}, P_{2} with fixed tangent lines. Then a direct computation proves the assertion (see section 3.2).

Assume that n < g/2. Since in general C · R_{n−1} = 3n − g + 1 then if n = (g − 1)/3, we get P_{1}, P_{2} ∉ R_{n−1}. Consider the linear system U_{a_{1},a_{2}} generated by

{ C ∈ U | C · f_{i} = a_{i}P_{i} + other points P ≠ P_{i}, i = 1, 2 }.

Then we have a chain Σ_{(P_{1},P_{2})} = U_{3,3} ⊆ U_{3,2} ⊆ U_{3,1} ⊆ U_{3,0} ⊆ U_{2,0} ⊆ U_{1,0} ⊆ U_{0,0} = |3H + (4 − g)f|. It suffices to check that all the inclusions are strict. To this purpose let R′, R′′, R′′′ ∈ |R_{g−n−1}| be irreducible and such that P_{1}, P_{2} ∉ R′, P_{1} ∈ R′′, P_{2} ∉ R′′, P_{1}, P_{2} ∈ R′′′. Then

(3 − a_{1})R′ + (a_{1} − a_{2})R′′ + a_{2}R′′′ ∈ U_{a_{1},a_{2}} \ ∪_{(b_{1},b_{2})>(a_{1},a_{2})}U_{b_{1},b_{2}}.

Here (b_{1}, b_{2}) > (a_{1}, a_{2}) means that (b_{1} − a_{1}, b_{2} − a_{2}) has non negative entries and at least one of them is positive.

Finally let (g − 1)/3 < n < g/2. Recall that the map

H^{0}(S, O_{S}(3H + (4 − g)f)) ⟶^{res} H^{0}(f_{1}, O_{f_{1}}(3))

is surjective from lemma 15 of [Co2]. If α = 1 this suffices to prove that dim(Σ_{P}) = 2g + 4 for P ∈ f_{1}.

Assume that α = 2. Let V := |res^{−1}(⟨v⟩)| where v ∈ H^{0}(f_{1}, O_{f_{1}}(3)) and div(v) := 3P_{1}. The surjectivity of res yields dim(V) = 2g+4. In order to prove that dim(Σ_{(P_{1},P_{2})}) = 2g+1 it suffices to prove that the restriction of V to f_{2} is |O_{f_{2}}(3)|.

If P_{1} ∉ R_{n−1}, for each P ∉ R_{n−1}, the set

A_{P} := { R ∈ |R_{g−n−1}| | P, P_{1} ∈ R }

contains an irreducible element. Indeed if R would be reducible, then R ∈ R_{n−1} + f_{1} + f_{2} + |(g − 2n − 2)f| which has dimension g − 2n − 2 < dim(A_{P}) = g − 2n − 1.

Now let P′ + P′′ + P′′′ ∈ |O_{f_{2}}(3)|, P′, P′′, P′′′ ∉ R_{n−1}. Choose irreducible elements R′ ∈ A_{P′}, R′′ ∈ A_{P′′}, R′′′ ∈ A_{P′′′}, and set C := R′ + R′′ + R′′′ + (3n − g + 1)f, f ≠ f_{2}. Then C · f_{2} = P′ + P′′ + P′′′. It follows that the restriction of V to f_{2} contains an open subset of |O_{f_{2}}(3)| hence they coincide.

If P_{1} ∈ R_{n−1} but P_{2} ∉ R_{n−1}, we can interchange their roles and repeat the above reasonment.

If P_{1}, P_{2} ∈ R_{n−1} then C · R_{n−1} = 3n − g + 1 ≥ 2. Consider the linear system V_{a} generated by

{ C ∈ V | C · f_{2} = aP_{2} + other points P ≠ P_{2} }.

Then we have a chain Σ_{(P_{1},P_{2})} = V_{3} ⊆ V_{2} ⊆ V_{1} ⊆ V_{0} = V. It suffices to check that all the inclusions are strict.

To this purpose, choose an irreducible R ∈ |R_{g−n−1} + f|. We have R ∩ R_{n−1} = P, and there is φ ∈ Aut(S) ≅ Aut(𝔽_{g−2n}) ≅ P_{n} ⋊ PGL_{2}, (P_{n} := Aut(O_{ℙ_{ℂ}^{1}} ⊕ O_{ℙ_{ℂ}^{1}}(2n − g))/ℂ^{∗}) such that φ(P) = P_{1}, thus we can always assume that P_{1} ∈ R, hence P_{2} ∉ R. It follows that

aR_{n−1} + (3 − a)R + (a(g − 2n) + (3n − g + 1) + a − 3)f ∈ V_{a} \ V_{a+1}

if f ≠ f_{2}. □

We can finally prove the main result of this section.

Theorem 2.2. If g ≥ 4 and α ≤ 2 then 𝔗_{g,n}(α) is irreducible and unirational.

Proof. First we assume g ≥ 5. Since there exists a natural rational map m: Σ(X) −−→ 𝔐_{g}, and m(Σ(X)) = 𝔗_{g,n}(α) then the irreducibility follows by lemma 2.1.

Since Σ(X) ⊆ X × |3H + (4 − g)f| as a scroll then it is rational, thus 𝔗_{g,n}(α) is unirational.

If g = 4 then n = 1, 2, thus the statement will follow from propositions 3.1.1, 3.2.1, 5.2, 5.3 below. □

We notice that the case α = 1 and g ≥ 5 was proved in [Co3].

Remark 2.3. If |D| is of kind n and 3P ∈ |D|, then the gap sequence at P is one of the following (see [Co1]):

(1, 2, 4, 5, …, 3n − 2, 3n − 1, 3n + 1, 3n + 4, …, 3(g − n − 1) + 1), I) (1, 2, 4, 5, …, 3n − 2, 3n − 1, 3n + 2, 3n + 5, …, 3(g − n − 1) + 2). II)

Accordingly we call P a t.r. of type I or of type II. It is proved in proposition 6 of [Co2] that P is of type II if and only if P ∈ R_{n−1}.

We introduce the loci 𝔗_{g,n}(α, σ) := { [C] ∈ 𝔗_{g,n} | the g_{3}^{1} has at least σ t.r. on R_{n−1} }. Clearly 0 ≤ σ ≤ α and 𝔗_{g,n}(α) = ⋃_{σ} 𝔗_{g,n}(α, σ) and, with the notations of the proof of theorem 2.2 above, 𝔗_{g,n}(α, 1) (resp. 𝔗_{g,n}(α, 2)) is dominated via m by p^{−1}(P_{1} × f_{2}) (resp. p^{−1}(P_{1} × P_{2})), hence it is irreducible and unirational too.

3. The loci 𝔗_{g,g/2}(α), α = 1, 2

Assume that g = 2n is even and let C be a curve representing a point [C] ∈ 𝔗_{g,n}. Then its canonical model lies on a unique ruled surface 𝔽_{0} isomorphic to ℙ_{ℂ}^{1} × ℙ_{ℂ}^{1}. We fix the natural embedding of 𝔽_{0} in ℙ_{ℂ}^{3} as a smooth quadric Q.

3.1. 𝔗_{g,n}(1). We fix a point A ∈ Q and a line a ⊆ Q through A. Let a′ ⊆ Q be the other line through A. Let π_{A}: Q → ℙ_{ℂ}^{2} be the projection Q from A onto a plane ℙ_{ℂ}^{2}. Such a map is an isomorphism outside a ∪ a′, is not defined at A and maps a and a′ onto two points P, P′ ∈ ℙ_{ℂ}^{2} respectively. We can choose coordinates in ℙ_{ℂ}^{2} in such a way that P = [1, 0, 0] and P′ = [0, 1, 0].

Each point [C] ∈ 𝔗_{g,n}(1) can be represented by a curve with a flex at A with tangent a. We can assume that the bidegree of C is (n + 1, 3) so that deg(C) = n + 4 and the trigonal pencil is cut out by the first ruling of Q and the trigonal map is induced by the projection on the second factor. Each such curve is mapped onto a curve \overline{C} of degree n + 3 having a n–uple ordinary point at P′ and a cusp at P with cuspidal tangent line passing through P′. Let V be the space of polynomials of degree n + 3 representing plane curves satisfying the above conditions. Notice that dim(V) = 4n + 5.

Let C_{1}, C_{2} ⊆ Q be curves of bidegree (n + 1, 3) having a flex at A with a as inflectional tangent line. Each isomorphism among them induces a projectivity of the their canonical space sending 𝔽_{0} into itself, thus an automorphism of Q. On the other hand each automorphism of Q is induced by a projectivity of ℙ_{ℂ}^{3}, which must fix both A and a, thus it gives rise to a projectivity φ: ℙ_{ℂ}^{2} → ℙ_{ℂ}^{2} with φ(P) = P, φ(P′) = P′ and φ(\overline{C_{1}}) = \overline{C_{2}}.

Conversely if such a φ does exist then we obtain an isomorphism among C_{1} and C_{2} outside a ∪ a′. Since both the curves are smooth then C_{1} ≅ C_{2}.

If we choose coordinates in ℙ_{ℂ}^{2} in such a way that P = [1, 0, 0] and P′ = [0, 1, 0] each such φ is represented by a matrix of the form

( \begin{matrix}α & 0 & β \\ 0 & γ & δ \\ 0 & 0 & ε \end{matrix} ) ,

and we denote by G the subgroup of PGL_{3} represented by matrices of the above form.

Proposition 3.1.1. If g = 2n, then 𝔗_{g,n}(1) ≈ ℙ(V)/G is irreducible, rational and of dimension 2g.

Proof. 𝔗_{g,n}(1) is irreducible since it is dominated by ℙ(V).

The birational isomorphism 𝔗_{g,n}(1) ≈ ℙ(V)/G follows from the above description. Its rationality is a consequence of a theorem of Vinberg since G is triangular (see [Do], section 6 or [P–V], theorem 2.11). Finally dim(𝔗_{g,n}(1)) = dim(V) − dim(G) = 4n = 2g. □

3.2. 𝔗_{g,n}(2). Assume that the isomorphism Q ≅ ℙ_{ℂ}^{1} × ℙ_{ℂ}^{1} ⊆ ℙ_{ℂ}^{3} is given by the map

S([s_{0}, s_{1}], [t_{0}, t_{1}]) := [x_{0}, x_{1}, x_{2}, x_{3}] = [s_{0}t_{0}, s_{0}t_{1}, s_{1}t_{0}, s_{1}t_{1}],

so that Q has equation x_{0}x_{3}−x_{1}x_{2} = 0. The group Aut(Q) = (PGL_{2}×PGL_{2})⋊⟨μ⟩, where the two copies of PGL_{2} act independently on the two rulings and μ acts on Q interchanging the two rulings.

We fix points A, B ∈ Q and lines a, b ⊆ Q of the same ruling through A and B respectively. Up to a projective transformation of ℙ_{ℂ}^{3} leaving Q fixed we can assume that A = [1, 0, 0, 0] = S([1, 0], [1, 0]), B = [0, 0, 0, 1] = S([0, 1], [0, 1]) and that a = {x_{2} = x_{3} = 0} and b = {x_{0} = x_{1} = 0}.

For each d, we denote by V_{d}(y) the space of binary forms of degree d in the two coordinates y_{0}, y_{1}. Each curve of bidegree (n + 1, 3) corresponds to a bihomogeneous form i.e. to an element of V_{n+1,3} := V_{n+1}(s) ⊗ V_{3}(t) up to a multiplicative constant factor.

Each point [C] ∈ 𝔗_{g,n}(2) can be represented by a curve with flexes at A, B with tangents a, b. Let V ⊆ V_{n+1,3} be the subspace representing curves on Q satisfying the above conditions. Omitting ⊗ for sake of brevity, if u ∈ V then

u(s, t) = u_{0}s_{0}^{n+1}t_{1}^{3} + u_{1}s_{1}^{n+1}t_{0}^{3} + all terms containing both s_{0} and s_{1} .

Assume that C_{1}, C_{2} ⊆ Q represent the same point of 𝔗_{g,n}(2). Then there exists a projective transformation φ: Q → Q sending C_{1} to C_{2} and fixing the set {A, B}. If φ(A) = A and φ(B) = B then φ is represented in Aut(Q) := (SL_{2} × SL_{2}) ⋊ ⟨μ⟩ by an element of

G_{0} := \{(( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) , ( \begin{matrix}β & 0 \\ 0 & β^{−1} \end{matrix} ) , id)\} ,

whereas if φ(A) = B and φ(B) = A, φ is represented by an element of

G_{1} := \{(( \begin{matrix}0 & −α \\ α^{−1} & 0 \end{matrix} ) , ( \begin{matrix}0 & −β \\ β^{−1} & 0 \end{matrix} ) , id)\} = G_{0}i,

where

i = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

acts on the two copies of SL_{2} by multiplication on the right.

In particular two elements of V represent isomorphic curves if and only if they differ by an element of G := G_{0} ⋊ ⟨i⟩.

Proposition 3.2.1. If g = 2n, then 𝔗_{g,n}(2) ≈ ℙ(V)/G is irreducible, rational and of dimension 2g − 1.

Proof. As in the proof of 3.1.1 𝔗_{g,n}(2) is irreducible since it is dominated by ℙ(V).

The birational isomorphism 𝔗_{g,n}(2) ≈ ℙ(V)/G follows from the above description. Let H := ℂ^{∗} × G where ℂ^{∗} acts on V by homotheties. Then 𝔗_{g,n}(2) ≈ V/H. We distinguish two cases according to the parity of n.

If n is even, say n = 2m, let

V′ := { u_{2}s_{0}^{m+1}s_{1}^{m}t_{0}^{2}t_{1} + u_{3}s_{0}^{m+1}s_{1}^{m}t_{0}t_{1}^{2} + u_{4}s_{0}^{m+1}s_{1}^{m}t_{0}^{3}+
+ u_{5}s_{0}^{m}s_{1}^{m+1}t_{1}^{3} + u_{6}s_{0}^{m}s_{1}^{m+1}t_{0}^{2}t_{1} + u_{7}s_{0}^{m}s_{1}^{m+1}t_{0}t_{1}^{2} ∈ V }.

Since H is reductive then V = V′ ⊕ V′′ as a representation of H. One can check that if u ∈ V′ is general enough then its stabilizer in H is

H_{u} = ⟨(−1, ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , id) , (−1, ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) , id)⟩ .

It is not difficult to check that H_{u} acts trivially on V.

If we prove that the quotient V′/H is rational, then the statement follows from section 4 of [Do]. Let

(3.2.1.1)  h := (γ, ( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) , ( \begin{matrix}β & 0 \\ 0 & β^{−1} \end{matrix} ) , id) ∈ H.

Then the condition

(3.2.1.2)  h(u_{2}^{λ_{2}}u_{3}^{λ_{3}}u_{4}^{λ_{4}}u_{5}^{λ_{5}}u_{6}^{λ_{6}}u_{7}^{λ_{7}}) = u_{2}^{λ_{2}}u_{3}^{λ_{3}}u_{4}^{λ_{4}}u_{5}^{λ_{5}}u_{6}^{λ_{6}}u_{7}^{λ_{7}}

yields a system of linear equations whose matrix is

A := ( \begin{matrix}1 & 1 & 1 & −1 & −1 & −1 \\ 1 & −1 & 3 & −3 & 1 & −1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix} ) .

If

B := ( \begin{matrix}1 & −1 & −2 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & −1 & 0 & −2 & −1 & 1 \end{matrix} )

then

AB = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 & 2 & −1 \\ 1 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) .

It follows that ℂ(u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7})^{H} = ℂ(X, S, T)^{⟨i⟩} where S = u_{3}u_{4}/u_{2}^{2}, T := u_{5}u_{6}/u_{7}^{2}, X := u_{3}u_{6}/u_{2}u_{7}. Since i(S) = T, i(T) = S, i(X) = X we finally have

ℂ(u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7})^{H} = ℂ(X, S, T)^{⟨i⟩} = ℂ(X, Y, Z)

where Y := S + T and Z = ST.

If n is odd, say n = 2m + 1, let

V′ := { u_{2}s_{0}^{m+1}s_{1}^{m+1}t_{0}^{3} + u_{3}s_{0}^{m+1}s_{1}^{m+1}t_{1}^{3} + u_{4}s_{0}^{m+1}s_{1}^{m+1}t_{0}^{2}t_{1}+
+ u_{5}s_{0}^{m+1}s_{1}^{m+1}t_{0}t_{1}^{2} + u_{6}s_{0}^{m+2}s_{1}^{m}t_{0}^{3} + u_{7}s_{0}^{m}s_{1}^{m+2}t_{1}^{3} ∈ V }.

Again V = V′ ⊕ V′′ and the stabilizer H_{u} in H of each general u ∈ V′ acts trivially on V.

We now prove that V′/H is rational. Let h ∈ H of the form (3.2.1.1). Then the condition (3.2.1.2) yields a system of linear equations whose matrix is

A := ( \begin{matrix}0 & 0 & 0 & 0 & 2 & −2 \\ 3 & −3 & 1 & −1 & 3 & −3 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix} ) .

Let

B := ( \begin{matrix}1 & 1 & 0 & 1 & −1 & −1 \\ 0 & 1 & 0 & 0 & 0 & −1 \\ −1 & −1 & 1 & −2 & 0 & 0 \\ 0 & −1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) .

Then

AB = ( \begin{matrix}0 & 0 & 0 & 0 & 2 & 0 \\ 2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} )

hence ℂ(u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7})^{H} = ℂ(X, Y, S)^{⟨i⟩} where S = u_{2}u_{5}/u_{4}^{2}, X := u_{2}u_{3}/u_{4}u_{5}, Y := u_{6}u_{7}/u_{2}u_{3}. If we set Z := S + X/S, since i(S) = X/S, then we have extensions ℂ(X, Y, Z) ⊆ ℂ(X, Y, S)^{⟨i⟩} ⊆ ℂ(X, Y, S). Moreover [ℂ(X, Y, S) : ℂ(X, Y, Z)] = 2 and ℂ(X, Y, S)^{⟨i⟩} ≠ ℂ(X, Y, S), thus we finally have

ℂ(u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7})^{H} = ℂ(X, S, S)^{⟨i⟩} = ℂ(X, Y, Z).

Finally dim(𝔗_{g,n}(2)) = dim(V) − dim(H) = 4n − 1 = 2g − 1. □

4. The loci 𝔗_{g,(g−1)/2}(α), α = 1, 2

Assume that g = 2n + 1 is odd and let C be a curve representing a point [C] ∈ 𝔗_{g,n}. Then its canonical model lies on the Hirzebruch surface 𝔽_{1} which coincides with ℙ_{ℂ}^{2} blown up at a point P ∈ ℙ_{ℂ}^{2}. Let p: 𝔽_{1} → ℙ_{ℂ}^{2} be the blow up morphism contracting the unique (−1)–section C_{∞} of 𝔽_{1} which is induced by |C_{∞} + f| (f is any fibre of the morphism 𝔽_{1} → ℙ_{ℂ}^{1}).

By the adjunction formula on 𝔽_{1}, we obtain C ∼ 3C_{∞} + (g + 5)/2f in Div(𝔽_{1}). Since

deg(C ∩ (C_{∞} + f)) = (g + 5)/2, deg(C ∩ C_{∞}) = (g − 1)/2,

p(C) := \overline{C} ⊆ ℙ_{ℂ}^{2} is a curve of degree n = (g + 5)/2 with a (g − 1)/2–uple point at P.

Notice that each point of total ramification P′ of C gives rise on \overline{C} ⊆ ℙ_{ℂ}^{2} to an ordinary flex at p(P′), whose tangent line passes through the singular point P and conversely. With a proper choice of the coordinates in ℙ_{ℂ}^{2} we can assume that P = [0, 0, 1], hence the equation of \overline{C} is

(4.1)  A_{n−3}(x_{0}, x_{1})x_{2}^{3} + B_{n−2}(x_{0}, x_{1})x_{2}^{2} + C_{n−1}(x_{0}, x_{1})x_{2} + D_{n}(x_{0}, x_{1}) = 0

where A_{n−3} := ∑_{i=0}^{n−3} a_{i}x_{0}^{i}x_{1}^{n−3−i}, B_{n−2} = …, … are binary forms of degree n−3, n−2, … respectively and n = (g + 5)/2 as above.

4.2. 𝔗_{g,n}(1). If [C] ∈ 𝔗_{g,n}(1) then \overline{C} has exactly one ordinary flex P′ whose tangent line contains P. Assume that P′ = [0, 1, 0]. C_{1}, C_{2} ⊆ 𝔽_{1} represent the same point of 𝔗_{g,n}(1) then if and only if there exists a projective transformation φ: ℙ_{ℂ}^{2} → ℙ_{ℂ}^{2} sending \overline{C_{1}} to \overline{C_{2}} and such that φ(P) = P, φ(P′) = φ(P′) (imitate the analogous proof on plame models of curves lying on 𝔽_{0}), hence φ is represented by a matrix of the form

( \begin{matrix}α & 0 & β \\ 0 & γ & δ \\ 0 & 0 & ε \end{matrix} ) .

Let G be the subgroup of PGL_{3} represented by matrices of the above form. Notice that a polynomial of the form (4.1) represent a curve with a flex at P′ if and only if b_{0} = c_{0} = d_{0} = 0. Let

V := { polynomials of the form (4.1), such that b_{0} = c_{0} = d_{0} = 0 }.

Proposition 4.2.1. If g = 2n + 1, then 𝔗_{g,n}(1) ≈ ℙ(V)/G is irreducible, rational and of dimension 2g.

Proof. As in the proof of proposition 3.1.1 it is a consequence of [Do], section 6 or [P–V], theorem 2.11. □

4.3. 𝔗_{g,n}(2). If [C] ∈ 𝔗_{g,n}(2) then \overline{C} has exactly two ordinary flexes P′, P′′ whose tangent lines contain P. Assume that P′ = [0, 1, 0], P′′ = [1, 0, 0]. If C_{1}, C_{2} ⊆ 𝔽_{1} represent the same point of 𝔗_{g,n}(2) then there exists a projective transformation φ: ℙ_{ℂ}^{2} → ℙ_{ℂ}^{2} sending \overline{C_{1}} to \overline{C_{2}}, such that φ(P) = P, and sending the set \{P′, P′′\} into itself.

If φ(P′) = P′ and φ(P′′) = P′′ then φ is represented by an element of

G_{0} := \{( \begin{matrix}α & 0 & 0 \\ 0 & β & 0 \\ 0 & 0 & γ \end{matrix} )\} ,

whereas if φ(P′) = P′′ and φ(P′′) = P′, φ is represented by an element of

G_{1} := \{( \begin{matrix}0 & α & 0 \\ β & 0 & 0 \\ 0 & 0 & γ \end{matrix} )\} = G_{0}i,

where

i = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Again two elements of V represent isomorphic curves if and only if they differ by an element of G := G_{0} ⋊ ⟨i⟩.

As in the previous case a polynomial of the form (4.1) represents a curve with a flex at P′ if and only if b_{0} = c_{0} = d_{0} = b_{n−2} = c_{n−1} = d_{n} = 0. Let

V := { polynomials of the form (4.1), such that b_{0} = c_{0} = d_{0} = b_{n−2} = c_{n−1} = d_{n} = 0 }.

Proposition 4.3.1. If g = 2n + 1, then 𝔗_{g,n}(2) ≈ ℙ(V)/G is irreducible, rational and of dimension 2g − 1.

Proof. For proving the statement about irreducibility and dimension imitate the analogous proofs in section 3.

The birational isomorphism 𝔗_{g,n}(2) ≈ ℙ(V)/G follows from the above description. Since the subgroup of scalar matrices acts trivially on V, we also have 𝔗_{g,n}(2) ≈ V/G. We distinguish two cases according to the parity of n.

If n is odd, say n = 2m + 1, let

V′ := { b_{m−1}x_{0}^{m−1}x_{1}^{m}x_{2}^{2} + b_{m}x_{0}^{m}x_{1}^{m−1}x_{2}^{2}+
+ c_{m}x_{0}^{m}x_{1}^{m}x_{2} + d_{m}x_{0}^{m}x_{1}^{m+1} + d_{m+1}x_{0}^{m+1}x_{1}^{m} ∈ V }.

Since G is reductive then V = V′ ⊕ V′′ as a representation of G. One can check that if u ∈ V′ is general enough then its stabilizer in G is

G_{u} = \{( \begin{matrix}ω & 0 & 0 \\ 0 & ω & 0 \\ 0 & 0 & ω \end{matrix} ) | ω^{n} = 1\} ,

which acts trivially on V. Since dim(V′/G) = 2 then V′/G is rational by a theorem of Castelnuovo. Now we can apply the results of section 4 of [Do], obtaining the rationality of V/G.

If n is even, say n = 2m, let

V′ := { a_{m−2}x_{0}^{m−2}x_{1}^{m−1}x_{2}^{3} + a_{m−1}x_{0}^{m−1}x_{1}^{m−2}x_{2}^{3}+
+ b_{m−1}x_{0}^{m−1}x_{1}^{m−1}x_{2}^{2} + c_{m−1}x_{0}^{m−1}x_{1}^{m}x_{2} + c_{m}x_{0}^{m}x_{1}^{m−1}x_{2} ∈ V }.

Since G is reductive then V = V′ ⊕ V′′ as a representation of G and, as above, for each general u ∈ V′ its stabilizer G_{u} in G is finite and it acts trivially on V. Since dim(V′/G) = 2 then V′/G is again rational. □

5. An application to A(𝔐_{4})

Let 𝔐_{4}^{(1)} be the locus of curves carrying a vanishing theta–null and ℌ_{4} the hyperelliptic locus. In [Fa] it is proved that ℌ_{4}, 𝔐_{4}^{(1)}, 𝔗_{4,2}(1), 𝔗_{4,2}(2), 𝔗_{4,1}(1) and 𝔗_{4,1}(2) are generators of the Chow ring A(𝔐_{4}) of 𝔐_{4}. For the rationality of ℌ_{4} see [P–V], theorem 2.12 and the references cited there, whereas the rationality of 𝔐_{4}^{(1)} is proved in [Do]. In section 2 we have proved the rationality of 𝔗_{4,2}(1) and 𝔗_{4,2}(2).

The aim of the present section is to verify that also 𝔗_{4,1}(1) and 𝔗_{4,1}(2) are rational varieties. Assuming this we obtain the following

Theorem 5.1. The generators of the Chow ring A(𝔐_{4}) described in [Fa] are rational. □

If C represents a point in 𝔗_{4,1}(α) then its canonical model lies on 𝔽_{1}. Blowing down we can embed C in ℙ_{ℂ}^{3} as a curve on a quadric cone Q not passing through the vertex. Up to a projective transformation in ℙ_{ℂ}^{3} the equation of Q is x_{1}^{2} − x_{0}x_{2} = 0.

Proposition 5.2. 𝔗_{4,1}(1) is irreducible and rational of dimension 7.

Proof. Let C ⊆ Q be a curve of genus g = 4 having a point A ∈ Q of total ramification. Let a ⊆ Q be the unique ruling of Q through A. The projection from A onto a plane H ⊆ ℙ_{ℂ}^{3} induces a birational morphism p: Q → H which is an isomorphism outside a sending a to a point P ∈ H. We can assume H = {x_{3} = 0}, A = [0, 0, 1, 0] so that a = {x_{0} = x_{1} = 0} and P := [0, 0, 1].

Each point [C] ∈ 𝔗_{4,1}(1) can be represented by a curve C with a point of total ramification at A. Its image via p is a curve \overline{C} of degree 5 with a double point at P, having a unique tangent line r = \{x_{0} = 0\} and no other singular points. Since \overline{C} has geometric genus g = 4 it follows that P is a point of type either A_{3} or A_{4}: on the other hand the plane H′ := \{x_{0} = 0\} intersects Q only along a, thus r ∩ \overline{C} = 5P, thus P is a point of type A_{4}.

It is easy to see that the equation of a curve of degree 5 having a point of type A_{4} at [0, 0, 1] with tangent line r = {x_{0} = 0} is of the form

(5.2.1)  f(x_{0}, x_{1}, x_{2}) = x_{0}^{2}x_{2}^{3} + x_{0}x_{2}^{2}f_{2}(x_{0}, x_{1}) + x_{0}x_{2}f_{3}(x_{0}, x_{1}) + f_{5}(x_{0}, x_{1}),

where f_{i}(x_{0}, x_{1}) are binary forms of degree i. Let V ⊆ ℂ[x_{0}, x_{1}, x_{2}] the space of forms of degree 5 of the form (5.2.1).

Assume that C_{1}, C_{2} ⊆ Q represent the same point of 𝔗_{4,1}(1). Then there exists a projective transformation φ: ℙ_{ℂ}^{2} → ℙ_{ℂ}^{2} sending \overline{C_{1}} to \overline{C_{2}}. Then φ(P) = P, φ(r) = r, thus each such φ is represented by a matrix of the form

( \begin{matrix}α & 0 & 0 \\ β & γ & 0 \\ δ & ε & ϑ \end{matrix} ) ,

Denote by G the subgroup of PGL_{3} represented by matrices of the above form. Then 𝔗_{4,1}(1) ≈ V/G which is rational since G is triangular (see [Do], section 6 or [P–V], theorem 2.11). □

Proposition 5.3. 𝔗_{4,1}(2) is irreducible and rational of dimension 6.

Proof. Let C ⊆ Q be a curve of genus g = 4 having two points A, B ∈ Q of total ramification. Let a, b ⊆ Q be the two rulings of Q through A, B respectively. We can assume A = [0, 0, 1, 0], B = [1, 0, 0, 0] so that a = {x_{0} = x_{1} = 0} and b = {x_{1} = x_{2} = 0}.

If φ ∈ PGL_{4} fixes Q then it leaves {A, B} and the vertex of Q fixed. If φ(A) = A and φ(B) = B then φ is represented by a matrix of the form

( \begin{matrix}α^{2} & a & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & c & β^{2} & 0 \\ 0 & δ & 0 & γ \end{matrix} ) .

By direct substitution one checks then that φ leaves the equation of Q unchanged if and only if a = c = 0 and b = αβ. Let

(5.3.2)  G_{0} = \{( \begin{matrix}α^{2} & 0 & 0 & 0 \\ 0 & αβ & 0 & 0 \\ 0 & 0 & β^{2} & 0 \\ 0 & δ & 0 & γ \end{matrix} )\} .

In the same way one checks that if φ(A) = B and φ(B) = A, then φ is represented by gi where g ∈ G_{0} and

i = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

Let G := G_{0} ⋊ ⟨i⟩.

Each curve C ⊆ Q of genus 4 is the complete intersection of Q and a cubic F. Since x_{1}^{2} − x_{0}x_{2} = 0 in the coordinate ring of C we can assume that F is the unique cubic in the ideal of C whose equation is

(5.3.3)  f_{3}(x_{0}, x_{2}, x_{3}) + x_{1}f_{2}(x_{0}, x_{2}, x_{3}) = 0

where f_{i} ∈ ℂ[x_{0}, x_{,}x_{3}] is a form of degree i. Since a and b are inflectional tangent lines at A and B respectively, we obtain that

f_{3}(x_{0}, x_{2}, x_{3}) = a_{1}x_{0}^{2}x_{2} + a_{2}x_{0}x_{2}^{2} + a_{3}x_{0}x_{2}x_{3} + a_{4}x_{3}^{3}.

Let V be the space of polynomials of the form (5.3.3) where f_{3} is as above. Then 𝔗_{4,1}(2) ≈ V/G.

Let

U := { a_{1}x_{0}^{2}x_{2} + a_{2}x_{0}x_{2}^{2} + a_{3}x_{0}x_{2}x_{3} + a_{4}x_{3}^{3}+
+ x_{1}(c_{0}x_{0}^{2} + c_{1}x_{2}^{2} + c_{2}x_{0}x_{2} + c_{3}x_{0}x_{3} + c_{4}x_{2}x_{3}) ∈ V }.

If g ∈ G_{0} is such that g(U) ⊆ U then δ = 0 in (5.3.2). Denoting by

H_{0} = \{( \begin{matrix}α^{2} & 0 & 0 & 0 \\ 0 & αβ & 0 & 0 \\ 0 & 0 & β^{2} & 0 \\ 0 & 0 & 0 & γ \end{matrix} )\}

and H = H_{0} ⋊ ⟨i⟩ one checks that U is a (G, H)–section of V, hence 𝔗_{4,1}(2) ≈ U/H.

Since H is reductive there exist an H–complement U′′ of the H–invariant subspace of U

U′ := { a_{3}x_{0}x_{2}x_{3} + x_{1}(c_{0}x_{0}^{2} + c_{1}x_{2}^{2} + c_{3}x_{0}x_{3} + c_{4}x_{2}x_{3}) ∈ V },

i.e. U = U′ ⊕U′′ as a representation of H. The stabilizer H_{u} in H of each general u ∈ U′ is finite, thus dim(U′/H) = 2 and U′/H is rational by a theorem of Castelnuovo. Moreover U/H is rational by section 4 of [Do]. □

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Gianfranco Casnati, Dipartimento di Matematica, Politecnico di Torino, c.so Duca degli Abruzzi, 10139 Torino, Italy

E-mail address: casnati@calvino.polito.it

Andrea Del Centina, Dipartimento di Matematica, Università degli Studi di Ferrara, via Machiavelli 35, 44100 Ferrara, Italy

E-mail address: cen@dns.unife.it