STABILITY COMPUTATION VIA GRÖBNER BASIS

BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

Contents

1. Introduction and Preliminaries 1 2. Hilbert-Mumford index of Hilbert points 3 2.1. A Gröbner basis algorithm for computing the Hilbert-Mumford

index 4

2.2. Macaulay 2 implementation 8 2.3. State polytopes 9 3. Applications 11 3.1. The rational bicuspidal curve of genus two 11 3.2. Degeneration of cuspidal curves 14 3.3. Hilbert unstable curves - Instability of elliptic bridges 18 References 25

1. Introduction and Preliminaries

In this article, we discuss a Gröbner basis algorithm related to the stability of algebraic varieties in the sense of Geometric Invariant Theory. We implement the algorithm with Macaulay 2, and give some applications to the moduli theory of curves.

Given an algebraic group G acting on a projective variety X linearized by a line bundle L, the stability of a point x ∈ X can be determined by examining its stability with respect to one-parameter subgroups ρ : 𝔾_{m} ) G. For each ρ, we let ρ(α).x specialize to a point x^{⋆} ∈ X and look at the character with which G acts on the fibre L_{x^{⋆}}: If the character is negative (resp. positive, nonnegative), then x and x^{⋆} are stable (resp. unstable, semistable) with respect to ρ. The negative of this character is called the Hilbert-Mumford index μ^{L}(x, ρ) of x with

Date: April 1, 2009.

2 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

respect to ρ. Assuming L is very ample, X is a closed subvariety of ℙ^{N} := ℙ(Γ(L)) and the Hilbert-Mumford index of x with respect to ρ admits the following simple description:

μ^{L}(x, ρ) = −min{wt_{ρ}(x_{i}) | x_{i}(x) ≠ 0}

where x_{i}’s are homogeneous coordinates of ℙ^{N} that diagonalize the action of ρ.

While computing the Hilbert-Mumford index of a given point in a projective space is simple and does not require an algorithm, this becomes quite a daunting task if the ‘point’ is itself complicated, sitting inside a large projective space. Our object of study in this paper is the prime example: In many moduli problems, algebraic geometers use the Hilbert scheme that parametrizes subschemes, and describing its points is not suitable for manual computation even for relatively simple subvarieties of a projective space of reasonable size: for instance, describing Hilbert points of genus two, degree six curves in ℙ^{4} using degree two generators would require 1365 variables!

The main algorithm in this paper uses Gröbner bases to effectively compute the Hilbert-Mumford index of the Hilbert point of a variety. The algorithm is implemented with Macaulay 2 in §2.2: Interested readers are invited to copy and paste the code mumfordIndex and use it to verify our computations or to carry out other stability computations. The Macaulay 2 script is available at

http://www.science.marshall.edu/hyeond.

As an application, we use the algorithms to prove the stability (with respect to a ρ) of certain curves of genus two with cusps, which play an important role in the geometry of the moduli space of tri-canonical curves [HL07b]. We also prove the instability of the bicanonical elliptic bridges (Definition 5), which was used in working out the GIT of bi-canonical curves [HL09, Proposition 10].

Acknowledgement. We thank Ian Morrison for bringing our attention to [BM88], and Hyungju Park and Dan Grayson for helping us with Macaulay 2 computations.

B.H. was partially supported by National Science Foundation Grants 0134259 and 0554491. D.H. was partially supported by Korea Institute for Advanced Study. D.H. and Y.L were supported by the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea government(MOST) (No. R01-2007-000-10948-0). Part of this article was completed while Y.L. was visiting RIMS at Kyoto University by the JSPS Invitation Fellowship Program. He thanks Shigefumi Mori for the invitation and hospitality during his stay at RIMS.

2. Hilbert-Mumford index of Hilbert points

Let X ⊂ ℙ^{N} = ℙ(V) be a projective variety with Hilbert polynomial P and Hilb, the component containing X of the Hilbert scheme parametrizing the subschemes of ℙ^{N} that have Hilbert polynomial P. We let [X] denote the Hilbert point of X in Hilb.

For m ≫ 0, we have a projective embedding

(2.1)  ϕ_{m} : Hilb ) Gr(P(m), Sym^{m}V^{∗}) ) ℙ := ℙ ( ⋀^{P(m)} Sym^{m}V^{∗}) .

The mth Hilbert point [X]_{m} of X is the image ϕ_{m}([X]) in ℙ.

Let x_{0}, …, x_{N} be homogeneous coordinates for ℙ^{N}. Let B_{m} denote the monomial basis {x^{a} = ∏ _{i=0}^{N} x_{i}^{a_{i}} | ∑ a_{i} = m} of Sym^{m}V^{∗}. The exterior products

x^{a(1)} ∧ x^{a(2)} ∧ ⋯ ∧ x^{a(P(m))} , x^{a(i)} ∈ B_{m}

form a basis W_{m} for ⋀^{P(m)} Sym^{m}V^{∗}.

Let ρ" : 𝔾_{m} ) SL_{N+1}(k) be a one-parameter subgroup. If x_{0}, …, x_{N} diagonalize the action of ρ", that is,

ρ"(t) · x_{i} = t^{r′_{i}}x_{i}, r′_{0} ≥ ⋯ ≥ r′_{N}, ∑ r′_{i} = 0,

then the bases B_{m} and W_{m} diagonalizes the action of ρ" on Sym^{m}V^{∗} and ⋀^{P(m)} Sym^{m}V^{∗}: For x^{a} := ∏ _{i=0}^{N} x_{i}^{a_{i}} , we have

ρ"(t) · x^{a} = t^{wt_{ρ"} (x^{a} )} x^{a}, wt_{ρ"}(x^{a}) = ∑_{i=0}^{N} r_{i}a_{i}.

For M = x^{a(1)} ∧ x^{a(2)} ∧ ⋯ ∧ x^{a(P(m))} ,

ρ"(t) · M = t^{wt_{ρ"} (M)} M, wt_{ρ"}(M) = ∑_{j=1}^{P(m)} wt_{ρ"}(x^{a(j)} ).

By definition, the Hilbert-Mumford index is

(2.2)  μ([X]_{m}, ρ") = max{−wt_{ρ"}(M) | M ≠ 0 on [X]_{m} }.

Let ρ : 𝔾_{m} ) GL_{N+1}(k) be the associated one-parameter subgroup with weights r_{i} = r′_{i}−r′_{N} such that r_{0} ≥ r_{1} ≥ ⋯ ≥ r_{N} = 0 and r′_{i} = r_{i} − \frac{1}{N+1} ∑ r_{j}. In practice,

4 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

we frequently start with a 1-PS ρ of GL_{N+1}(k) with weight r_{i} and compute the Hilbert-Mumford index with respect to the 1-PS ρ′ of SL_{N+1}(k) with integral weights (N + 1)r_{i} − ∑ r_{j}. Since ρ′ = (N + 1)ρ", the (semi)stability with respect to ρ′ is equivalent to the (semi)stability with respect to ρ".

Given M ∈ W_{m}, the ρ-weight and the ρ′-weight of M are related by

(2.3)  wt_{ρ′}(M) = (N + 1)wt_{ρ}(M) − r · m · P(m)

where r = ∑ _{i=0}^{N} r_{i}. Combining (2.2) and (2.3), we obtain

(2.4) μ([X]_{m}, ρ′) = (N + 1) · max{−wt_{ρ}(M) | M ≠ 0 on [X]_{m} } + r · m · P(m).

For notational convenience, we define μ([X]_{m}, ρ) := μ([X]_{m}, ρ′).

2.1. A Gröbner basis algorithm for computing the Hilbert-Mumford index. This algorithm seems to have been known to certain experts (see [Bay82] and [BM88]). Indeed, Lemma 3.3 and Corollary 3.4 of [BM88] deal with the generic case of our Proposition 1. We write the details here in a form convenient for our application.

Given a one-parameter subgroup ρ of GL_{N+1}(k) such that ρ(α) · x_{i} = α^{r_{i}} x_{i}, r_{0} ≥ r_{1} ≥ ⋯ ≥ r_{N} = 0, introduce the following ρ-weighted graded lexicographic order, denoted simply by ‘≺’. This is a total order on the set of monomials {x^{a}} defined by declaring that x^{a} ≺ x^{b} if

(1) deg x^{a} < deg x^{b} or

(2) deg x^{a} = deg x^{b} and wt_{ρ}(x^{a}) < wt_{ρ}(x^{b}) or

(3) deg x^{a} = deg x^{b} and wt_{ρ}(x^{a}) = wt_{ρ}(x^{b}) and a_{j} < b_{j} where

j = min{i | a_{i} ≠ b_{i}}.

Given f ∈ S := k[x_{0}, …, x_{N}], we let in_{≺}(f) denote the term of f with maximal order. For an ideal I of S, we let

in_{≺}(I) := ⟨in_{≺}(f) | f ∈ I⟩.

Let I_{X} be the homogeneous ideal of a subscheme X ⊂ ℙ^{N}. Note that the monomials {x^{a(1)}, …, x^{a(P(m))} } that are not in in_{≺}(I_{X}) form a basis of (S/I_{X})_{m} and (S/in_{≺}(I_{X}))_{m}.

Proposition 1. The Hilbert-Mumford index of the mth Hilbert point of X with respect to the associated one-parameter subgroup ρ′ of SL_{N+1}(k) is:

(2.5)  μ([X]_{m}, ρ′) = −(N + 1) ∑_{i=1}^{P(m)} wt_{ρ}(x^{a(i)} ) + m · P(m) · ∑_{j=0}^{N} r_{j}

where {x^{a(1)}, …, x^{a(P(m))} } are degree m monomials not in in_{≺}(I_{X})

Proof. Let Z = {x^{b(1)}, …, x^{b(P(m))}} be another P(m)-element subset of B_{m} that gives rise to a basis for (S/I_{X})_{m}. Note that Z being a basis is equivalent to that x^{b(1)} ∧ ⋯ ∧ x^{b(P(m))} is nonzero on the Hilbert point [X]_{m}. Consider the normal form ∑ _{j=1}^{P(m)} c_{ij}x^{a(j)} of x^{b(i)} determined uniquely by

x^{b(i)} ≡ ∑_{j=1}^{P(m)} c_{ij}x^{a(j)} (mod I_{X}), c_{ij} ∈ k.

Since both {x^{a(1)}, …, x^{a(P(m))} } and Z are bases for the quotient space (S/I_{X})_{m}, the matrix (c_{ij}) is invertible. This allows us to reorder x^{b(i)}’s as follows: Determine τ_{1} by the condition that the τ_{1}th row contains the pivot element of the matrix (c_{ij}), where pivot is simply the first nonzero entry of the first column. Given new {b(τ_{1}), …, b(τ_{j−1})}, we define τ_{j} by the condition that the τ_{j}th row contains the pivot of the (P(m)−j+1)×(P(m)−j+1) matrix obtained from (c_{ij}) by deleting the rows and columns that contain the first j − 1 pivots. Our choice of τ_{i}’s insures that after the reordering, we have a one to one correspondence x^{a(i)} ) x^{b(τ_{i} )} between {x^{a(1)}, …, x^{a(P(m))} } and Z such that

x^{b(τ_{i} )} ≡ ∑_{k=1}^{P(m)} c′_{ik}x^{a(k)} (mod I_{X})

where c′_{ik} := c_{τ_{i} k} and c′_{ii} ≠ 0. It follows that

wt_{ρ} (in_{≺} (g_{i})) = wt_{ρ}(x^{b(τ_{i} )}) ≥ wt_{ρ}(x^{a(i)} ), g_{i} := x^{b(τ_{i} )} − ∑_{k=1}^{P(m)} c′_{ik}x^{a(k)} ∈ I_{X}.

Hence

∑_{i=1}^{P(m)} wt_{ρ}(x^{b(τ_{i} )}) ≥ ∑_{i=1}^{P(m)} wt_{ρ}(x^{a(i)} )

and the assertion follows from (2.4). □

The proposition translates into the following stability statements:

Corollary 1. [X]_{m} is stable (resp. semistable) with respect to ρ if and only if

∑ wt_{ρ}(x^{a(i)} ) < (resp. ≤)\frac{mP(m)}{N + 1} ∑ r_{i}.

In terms of the corresponding one-parameter subgroup ρ′ of SL_{N+1}(k),

Corollary 2. [X]_{m} is stable (resp. semistable) with respect to ρ′ if and only if

∑_{i=1}^{P(m)} wt_{ρ′}(x^{a(i)} ) < (resp. ≤)0.

The upshot of the formula (2.5) is that the monomials x^{a(1)}, …, x^{a(P(m))} can be systematically computed by using Gröbner basis and can easily be implemented with a computer algebra system.

Moreover, considering the functoriality of the Hilbert-Mumford index and the tautological ring of the Hilbert scheme reveals that one only needs to compute the Hilbert-Mumford index for mth Hilbert points for finitely many m to obtain the Hilbert-Mumford indices for all m. The results in the remainder of the section are taken from [HH07].

Proposition 2. Let X, ρ, {x^{a(1)}, …, x^{a(P(m))} } be as before. The filtered Hilbert function P_{X,ρ} on ℤ defined by

P_{X,ρ}(m) = ∑_{i=1}^{P(m)} wt_{ρ′}(x^{a(i)} )

is a polynomial in m for m ≫ 0.

Proof. For m ≫ 0, we have the Grothendieck embedding (2.1) such that ϕ_{m}^{∗}O(+1) = detπ_{∗}O_{X} (m) where π : X ) Hilb is the universal variety. Let n be the dimension of X. There are Cartier divisors ([KM76]) L_{0}, …, L_{n+1} on Hilb such that

detπ_{∗}O_{X} (m) = ∑_{i=0}^{n+1} \binom{m}{i}L_{i}

and it follows from the functoriality of the Hilbert-Mumford index that

μ^{ϕ_{m}^{∗} O(+1)} ([X]_{m}, ρ) = ∑_{i=0}^{n+1} \binom{m}{i}μ^{L_{i}}([X], ρ)

which is a polynomial in m. □

STABILITY COMPUTATION VIA GRÖBNER BASIS 7

When this is put into practice to actually compute μ([X]_{m}, ρ) for all m, one needs to somehow determine M for which P_{X,ρ}(m) is a polynomial for all m ≥ M. An obviously necessary condition is that the mth Hilbert point of X be defined, which leads us to the Castelnuovo-Mumford regularity:

Proposition 3. If X and lim_{t)0} ρ(t) · X are M-regular, then we have

P_{X,ρ}(m) = ∑_{i=0}^{n+1} \binom{m}{i}μ^{L_{i}}([X], ρ)

for m ≥ M.

A very useful corollary of this is

Corollary 3. Let C ⊂ ℙ(V) be a projective variety, ρ : 𝔾_{m} ) SL(V) a oneparameter subgroup, and C^{⋆}, the variety to which ρ(t).C specializes. Suppose that C and C^{⋆} satisfy

(1) C (resp. C^{⋆}) is connected of pure dimension one;

(2) V^{∗} ) Γ(O_{C}(1)) (resp. Γ(O_{C^{⋆}}(1))) is an isomorphism;

(3) O_{C} (resp. O_{C^{⋆}}) is 2-regular. Then for each m ≥ 2 we have (2.6)

μ([C]_{m}, ρ) = (m−1) ((\frac{1}{2}μ([C]_{3}, ρ) − μ([C]_{2}, ρ)) m + 3μ([C]_{2}, ρ) − μ([C]_{3}, ρ)) .

Proof. (1) and (3) together imply that μ([C]_{m}, ρ) is a polynomial in m for m ≥ 2. (2) implies that detπ_{∗}O_{X} (1) = L_{0} + L_{1} is trivial, and the formula (2.6) follows immediately. □

The conditions in the above corollary are satisfied by a large class of curves, including the c-semistable curves i.e. reduced complete connected curves C such that

• C has nodes, ordinary cusps and tacnodes as singularities;

• the dualizing sheaf ω_{C} is ample;

• C does not have a genus-one connected subcurve that meets the rest of the curve in one point not counting multiplicity.

Corollary 4. Let C be a bicanonical c-semistable curve i.e., a c-semistable curve embedded by the bicanonical system |ω_{C}^{⊗2}| :

C ) ℙΓ(C, ω_{C}^{⊗2}) ≃ ℙ(V).

8 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

Let C^{⋆} denote the curve to which ρ(t).C specializes. If C^{⋆} is also a bicanonical c-semistable curve, then for all m ≥ 2, C is

(1) m-Hilbert stable if and only if μ([C]_{3}, ρ) ≥ 2μ([C]_{2}, ρ) > 0;

(2) m-Hilbert strictly semistable if and only if μ([C]_{3}, ρ) = μ([C]_{2}, ρ) = 0;

(3) m-Hilbert unstable if and only if μ([C]_{3}, ρ) ≤ 2μ([C]_{2}, ρ) < 0.

2.2. Macaulay 2 implementation. Here we give a Macaulay 2 [GS] implementation of the algorithm according to Proposition 1. The code has Input: A homogeneous ideal I of a graded ring S and a weight vector w. Output: A sequence consisting of

(1) The regularity reg(I) of I;

(2) Values of the filtered Hilbert function P_{X,ρ}(m) for m < reg(I), where X is the projective variety defined by I and ρ is the 1-PS whose weight vector is w;

(3) The polynomial which coincides with P_{X,ρ}(m) for m ≥ reg(I).

Function:

mumfordIndex = (I,w) -> (
S = ring I;
r = dim Proj(S/I);
regI = regularity resolution I;
MUm = (I,w,m) -> (
S = ring I;
N = numgens S;
K = coefficientRing S;
Sw = K[gens S, Weights => w, MonomialOrder => GLex];
W = map(Sw, S, vars Sw);
I = W(I);
P = hilbertPolynomial I;
inI = ideal leadTerm I;
Sbar = Sw/inI;
F = map(Sbar, Sw, vars Sbar);
Bm = basis(m, Sw);
-- Computes a basis of the degree m piece of Sw.
Bmbar = basis(m, Sbar);

STABILITY COMPUTATION VIA GRÖBNER BASIS 9

-- Computes a basis of the degree m piece of Sbar.
Bm = flatten entries Bm;
PSm = #Bm;
monomialWeight = (f) ->
 (expf = flatten exponents f;
  sum(expf, w, times));
e = apply(0..(PSm-1),i->(if F(Bm_i)===F(0) then 0 else 1));
TOTALWT = sum for i from 0 to PSm-1 list
product{monomialWeight(Bm_i), e_i};
-- Computes the total weight.
mu = sum{product{N,-TOTALWT}, product{m, P(m), sum w}}
-- Computes the Hilbert-Mumford index.
);
b = transpose matrix(QQ,table(1,r+2,(i,j)->MUm(I,w,regI + j)));
A = matrix(QQ,table(r+2,r+2, (i,j) -> (regI + i)^j));
c = A^(-1)*b;
QQ[m];
fHilbFun = sum(r+2, i->c_(i,0)*m^i);
-- Computes the filtered Hilbert polynomial.
if regI > 2 then
val = apply(i = 2..(regI-1), i->MUm(I,w,i))
else val = ();
print(regI, val, fHilbFun)
)

Remark 1. The subprogram MUm computes μ([X]_{m}, ρ) for a given m. After running MUm, the initial ideal in_{≺}(I) and the monomial basis {x^{a(1)}, …, x^{a(P(m))} } for (S/I)_{m} can be retrieved with the commands inI and Bmbar, respectively.

2.3. State polytopes. In [BM88], Bayer and Morrison considered the weight polytope of the mth Hilbert point [I]_{m} := ⋀^{P(m)} Sym^{m}V^{∗}/I_{m}: For a fixed maximal torus H ⊂ SL_{N+1}(k), the weight polytope is simply the convex hull of the characters of H that appear in the weight decomposition. This is called the (mth) state polytope of I and is denoted by State_{m}(I). The main theorem of [BM88] says that the vertices of State_{m}(I) are precisely

∑_{x^{a} ∈(in_{≺}I)_{m}} a, ≺ a monomial order .

Let ρ be a 1-PS of SL_{N+1}(k) with weight vector w. It follows from the definition of State_{m}(I) that

μ([I]_{m}, ρ) = max{−w.v | v a vertex of State_{m}(I) }

and Proposition 1 says that the maximum is achieved precisely at the vertex associated to in_{≺_{w}} (I), where ≺_{w} is the w-weighted lexicographic total order on the monomials.

State polytopes have received deserved attention after the fundamental work [Bay82] and [BM88]. Especially of our interest is [KSZ92] which proves that the Chow polytope can be realized as a suitable limit of the state polytopes. The Chow polytope Chow(I) of an ideal I is the weight polytope (with respect to a maximal torus) of the Chow form Ch(I). The precise statement is

lim ; m) \frac{(n + 1)!}{m^{n+1}} State_{m}(I) = Chow(I)

where n is the dimension of the projective variety defined by I. Let w ∈ ℝ^{N+1} and consider the linear functional L_{w}(x) = −w.x. Since L_{w} achieves its maximum on State_{m}(I) at the vertex associated to in_{≺_{w}} (I)_{m}, its maximum on Chow(I) is achieved at

lim ; m) \frac{(n + 1)!}{m^{n+1}} ∑_{x^{a} ∈in_{≺w} (I)_{m}} a.

It follows that

Corollary 5. For any 1-PS ρ of SL_{N+1}(k), we have

lim ; m) \frac{(n + 1)!}{m^{n+1}} μ([I]_{m}, ρ) = μ(Ch(I), ρ).

In particular, if a projective variety is asymptotically Hilbert semistable, then it is Chow semistable.

STABILITY COMPUTATION VIA GRÖBNER BASIS 11

3. Applications

In this section, we shall give two concrete applications of the algorithm developed in the previous section. These examples played important roles in our work on moduli problems of ν-canonical curves for ν = 2, 3 [HL07a, HL07b, HL09, HH07].

3.1. The rational bicuspidal curve of genus two. When constructing a moduli space, one hopes to avoid objects with infinite automorphisms as the moduli space often fails to be separated at such points. Fortunately, such objects are often destablized by a one-parameter subgroups of the automorphism group. However, if the object is not destabilized by one of these subgroups, then it has a rather good chance of being semistable. In the moduli problem of tricanonical curves of genus two [HL07b], the rational curve C_{0} with two cusps and no other singularities turns out to be at the focal point of the whole problem: It is the only pseudo-stable curve ([Sch91]) with infinite automorphisms to which other cuspidal pseudo-stable curves specialize under the action of Aut(C_{0}) (Figure 1).

In this section, we test C_{0} against the one-parameter subgroups ρ coming from Aut(C_{0}) and show that it is Hilbert strictly semistable with respect to ρ. Then we prove in §3.2 that [C_{0}]_{m} is the flat limit of the families {ρ(α) · [C′]_{m}} where C′ is any other pseudo-stable cuspidal curve. This implies that all cuspidal curves are strictly semistable with respect to ρ, and that all such curves are semistable if one of them is. The results in this section appeared without computational details in [HL07b] where we proved that these curves are semistable using a standard degeneration argument.

We first find a normalization map for C_{0} from the classical projective geometric construction of a cusp. Let ν_{6}(ℙ^{1}) denote the rational sextic curve

\begin{matrix}ν_{6} : ℙ^{1} & ) & ℙ^{6} \\ [s, t] & ) & [s^{6}, s^{5}t, s^{4}t^{2}, s^{3}t^{3}, s^{2}t^{4}, st^{5}, t^{6}] . \end{matrix}

To create a cusp at ν_{6}(0) = [0, …, 0, 1], we project ν_{6}(ℙ^{1}) from [0, …, 0, 1, 0] on the tangent line T_{ν_{6}(0)}ν_{6}(ℙ^{1}) = {X_{0} = ⋯ = X_{4} = 0}. The image under this projection is

π_{[0,...,0,1,0]} ∘ ν_{6}(ℙ^{1}) = {[s^{6}, s^{5}t, s^{4}t^{2}, s^{3}t^{3}, s^{2}t^{4}, t^{6}] | [s, t] ∈ ℙ^{1}}.

12 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

We successively project C′ from [0, 1, 0, …, 0] ∈ T_{[1,0,...,0]}C′ = {X_{2} = ⋯ = X_{5} = 0} and get

C_{0} = {[s^{6}, s^{4}t^{2}, s^{3}t^{3}, s^{2}t^{4}, t^{6}] | [s, t] ∈ ℙ^{1}}

which has ordinary cusps at [0, …, 0, 1] and [1, 0, …, 0]. From this, it is clear that C_{0} admits automorphisms coming from automorphisms [s, t] ) [αs, t], α ∈ 𝔾_{m}, of ℙ^{1}. Such an automorphism corresponds to the one-parameter subgroup ρ of GL_{5}(k) with weights (6, 4, 3, 2, 0). We shall prove that [C_{0}]_{m} is semistable with respect to ρ, for all m, via an explicit computation of Hilbert-Mumford index μ([C_{0}]_{m}, ρ). Although this can be done by simply plugging the ideal of C_{0} and w = {6, 4, 3, 2, 0} in mumfordIndex (§2.2), we shall first carry out the algorithm step by step and present the computations in a traditional manner as if we did them by hand.

• We first find the ideal I_{C_{0}} of C_{0} from the parametrization map:

I_{C_{0}} = ⟨−x_{1}x_{4} + x_{3}^{2}, −x_{0}x_{4} + x_{1}x_{3}, −x_{0}x_{4} + x_{2}^{2}, −x_{0}x_{3} + x_{1}^{2}⟩.

• We compute a Gröbner basis for I_{C_{0}} with respect to the ρ-weighted GLex:

x_{1}x_{4} − x_{3}^{2}, x_{0}x_{4} − x_{2}^{2}, x_{1}x_{3} − x_{2}^{2}, x_{0}x_{3} − x_{1}^{2}, x_{2}^{2}x_{4} − x_{3}^{3}, x_{0}x_{2}^{2} − x_{1}^{3}

• The leading terms of the Gröbner basis elements are:

x_{1}x_{4}, x_{0}x_{4}, x_{1}x_{3}, x_{0}x_{3}, x_{2}^{2}x_{4}, x_{0}x_{2}^{2}

These generate the initial ideal in_{≺}(I_{C_{0}} ).

• The degree 2 monomials not in the initial ideal are:

(3.1)  x_{0}^{2}, x_{0}x_{1}, x_{0}x_{2}, x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}, x_{2}x_{3}, x_{2}x_{4}, x_{3}^{2}, x_{3}x_{4}, x_{4}^{2}

These monomials have total weight 66. On the other hand, we have

\frac{P(2) · 2}{5} ∑_{i=0}^{4} r_{i} = \frac{11 · 2}{5} · 15 = 66.

Therefore, by Proposition 1, the 2nd Hilbert point of the tri-canonical image of C_{0} is at best strictly semistable with respect to ρ. Similarly, we find that the degree three monomials not contained in the initial ideal in_{≺}(I_{C_{0}} ) are:

\begin{matrix}x_{0}^{3}, x_{0}^{2}x_{1}, x_{0}^{2}x_{2}, x_{0}x_{1}^{2}, x_{0}x_{1}x_{2}, x_{1}^{3}, x_{1}^{2}x_{2}, x_{1}x_{2}^{2}, x_{2}^{3}, \\ x_{2}^{2}x_{3}, x_{2}x_{3}^{2}, x_{2}x_{3}x_{4}, x_{2}x_{4}^{2}, x_{3}^{3}, x_{3}^{2}x_{4}, x_{3}x_{4}^{2}, x_{4}^{3} \end{matrix}

of which total weight is 153. On the other hand, we have

\frac{P(3) · 3}{5} · ∑_{i=0}^{4} r_{i} = \frac{17 · 3}{5} · 15 = 153.

Therefore, by Proposition 1, the 3rd Hilbert point of the tri-canonical image of C_{0} is strictly semistable with respect to ρ. Now it follows from Corollary 4 that C_{0} is m-Hilbert strictly semistable for all m ≥ 2.

Remark 2. (μ([C_{0}]_{m}, ρ) as computed by Macaulay 2) First, compute the ideal of C_{0}:

i12 : P1 = QQ[s,t];
i13 : P4 = QQ[x_0..x_4];
i14 : f = map(P1,P4,{s^6,s^4*t^2,s^3*t^3,s^2*t^4,t^6})
                  6   4 2   3 3   2 4   6
o14 = map(P1,P4,{s , s t , s t , s t , t })
o14 : RingMap P1 <--- P4
i15 : C0 = kernel f
              2                       2          2
o15 = ideal (x  - x x , x x  - x x , x  - x x , x  - x x )
              3    1 4   1 3    0 4   2    0 4   1    0 3
o15 : Ideal of P4

Run mumfordIndex to compute the filtered Hilbert function P_{C_{0}/ρ}(m):

i7 : mumfordIndex(C0, {6,4,3,2,0})
(2, (), 0)

Reading the output sequence, the regularity of C_{0} is 2 and the filtered Hilbert function P_{C_{0}/ρ}(m) agrees with the zero polynomial for all m ≥ 2 (hence the empty sequence () in the second entry). Thus C_{0} is m-Hilbert strictly semistable for all m ≥ 2.

We can run the subprogram MUm to find μ([C_{0}]_{m}, ρ) for m = 2, 3 and the monomials not in the initial ideal:

i8 : MUm(C0, {6,4,3,2,0}, 2)
o8 = 0
i11 : inI
                                     2       2
o11 = ideal (x x , x x , x x , x x , x x , x x )

14 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

              1 4   0 4   1 3   0 3   2 4   0 2
o11 : Ideal of Sw

The degree two monomials not in the initial ideal are

i12 : Bmbar
o12 = | x_0^2 x_0x_1 x_0x_2 x_1^2 x_1x_2 x_2^2 x_2x_3 x_2x_4
        x_3^2 x_3x_4 x_4^2 |
                 1          11
o12 : Matrix Sbar  <--- Sbar

whose weights sum up to

i13 : TOTALWT
o13 = 66

Similarly, we compute μ([C_{0}]_{3}, ρ) by

i14 : MUm(C0, {6,4,3,2,0}, 3)
o14 = 0

The degree 3 monomials not in the initial ideal are

i15 : Bmbar
o15 = | x_0^3 x_0^2x_1 x_0^2x_2 x_0x_1^2 x_0x_1x_2 x_1^3
        x_1^2x_2 x_1x_2^2 x_2^3 x_2^2x_3 x_2x_3^2 x_2x_3x_4
        x_2x_4^2 x_3^3 x_3^2x_4 x_3x_4^2 x_4^3 |
                 1          17
o15 : Matrix Sbar  <--- Sbar

and their weights sum up tp

i16 : TOTALWT
o16 = 153

3.2. Degeneration of cuspidal curves. There are three types of genus two pseudo-stable curves with a cusp:

(a) C_{0}, the rational curve with two cusps;

(b) C′_{0}, the rational curve with a cusp and a node;

(c) C_{1}, a curve with one cusp and no other singularities, normalized by a

smooth elliptic curve. In this section, we shall prove C′_{0} and C_{1} specialize to C_{0} under the ρ-action. Since C′_{0} is in the closure of the locus of C_{1} in the Hilbert scheme, we only need to prove it for C_{1}.

Figure 1. C_{1} and C′_{0} degenerate to C_{0} along the action of ρ

3.2.1. Flat limit of ρ(α).[C_{1}]. Computation of flat limits is rather well known (cf. [BM88]). We quickly recapitulate the algorithm here: Given S = k[x_{0}, …, x_{N}] and a one-parameter subgroup ρ : 𝔾_{m} ) GL_{N+1}(k) defined by ρ(α).x_{i} = α^{wt_{ρ} (x_{i} )} x_{i}, we define the graded ρ-weight order ≺_{ρ} as follows: let x^{a} and x^{b} be monomials. Then x^{a} ≺_{ρ} x^{b} if

(1) deg x^{a} < deg x^{b} or

(2) deg x^{a} = deg x^{b} and wt_{ρ}(x^{a}) < wt_{ρ}(x^{b}).

Note that ≺_{ρ} is a partial order and ≺ in §2.1 is a total order that refines ≺_{ρ}.

Given g = ∑ c_{a}x^{a} ∈ S, and a homogeneous ideal I of S, we define

(a) in_{≺_{ρ}}(g) is the sum of the terms (of g) of maximal order^{1};

(b) in_{≺_{ρ}}(I) := ⟨in_{≺_{ρ}}(f) | f ∈ I⟩.

(c) g̃(x_{0}, …, x_{N}, α) := α^{b}g(α^{−r_{0}} x_{0}, …, α^{−r_{N}} x_{N}), b = max{wt_{ρ}(x^{a}) | c_{a} ≠ 0};

(d) Ĩ:= ⟨g̃, g ∈ I⟩ ⊂ S[α].

1Bayer and Mumford take the minimal weight term to be the initial term. Here we are using the dual action of ρ on the ideal, hence the reversal of the signs of the weights.

16 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

Note that for a fixed α ≠ 0, I_{α} = ⟨g̃| g ∈ I⟩ is the ideal defining ρ(α).C where C ⊂ ℙ^{N} is the projective variety defined by I. The problem at hand is to compute the ideal of the limit variety of the family ρ(α).C as α tends to zero. First, we have:

Theorem 3. ([Eis95] P.343) For any ideal I ⊂ S, the k[α]-algebra S[α]/Ĩ is free as k[α]-algebra. Furthermore, we have

\begin{matrix}S[α]/Ĩ⊗_{k[α]} k[α, α^{−1}] ≃ (S/I)[α, α^{−1}] \\ S[α]/Ĩ⊗_{k[α]} k[α]/(α) ≃ S/in_{≺_{ρ}}(I). \end{matrix}

This means precisely that Ĩ is the homogeneous ideal of the flat projective closure in ℙ^{N} of the family ρ(α).C, and that the flat limit is given by the initial ideal in_{≺_{ρ}}(I). These ideals can be readily computed by using Gröbner basis:

Proposition 4. ([Eis95] P.369) Let {g_{1}, …, g_{t}} be a Gröbner basis for I with respect to ≺ (§2.1). Then

(1) g̃_{1}, …, g̃_{t} generate Ĩ;

(2) in_{≺_{ρ}}(g_{1}), …, in_{≺_{ρ}}(g_{t}) generate in_{≺_{ρ}}(I).

Remark 4. This algorithm can be easily implemented with Macaulay 2. The following function flatLimit takes an ideal I and a weight vector w, and computes the projective closure tI and the flat limit of the one-parameter family ρ(α).I, α ∈ ℂ^{∗}, where ρ is the one-parameter subgroup with the prescribed weight vector w. The Gröbner basis computation occurs in saturate(I,a).

flatLimit = (I,w) -> (
R = ring I;
N = numgens R - 1;
K = coefficientRing R;
Ra = K[gens R, a];
wmax = max w;
f = map(Ra, R, gens ideal apply(0..N, j -> a^(wmax-w_j)*(gens Ra)_j));
-- the new weights wmax-w_j corresponds to those of the inverse of lambda
I = f(I);
tI = saturate(I,a);
substitute(tI, {a=>0})
)

Using this algorithm, we shall prove that lim_{α)0} ρ(α).C_{1} = C_{0}.

STABILITY COMPUTATION VIA GRÖBNER BASIS 17

(A) We first compute the ideal of C_{1}. Let ν : C_{1}^{ν} ) C_{1} be the normalization of C_{1} and p ∈ C_{1}^{ν} be the closed point over the cusp q of C_{1}. The dualizing sheaf ω_{C_{1}} can be expressed

ω_{C_{1}} (U) = { ζ ∈ ω_{C_{1}^{ν}} (ν^{−1}U) | ∑_{y∈ν^{−1} (x)} Res_{y}(ν^{∗}f · ζ) = 0 for all x ∈ U and f ∈ O_{C_{1},x} }

to an open set U ⊂ C_{1}. It follows that ν^{∗}ω_{C_{1}} = ω_{C_{1}^{ν}} (2p) = O_{C_{1}^{ν}} (2p) and ν^{∗}O_{C_{1}} (1) ≃ O_{C_{1}^{ν}} (6p). This means that the tri-canonical image of C_{1} is given by a 𝔤_{6}^{4} of C_{1}^{ν}. In other words, C_{1} is the image of a suitable projection

ℙ(Γ(C_{1}^{ν}, O_{C_{1}^{ν}} (6p))) −−→ ℙ^{4}

following the embedding η : C_{1}^{ν} ) ℙ(Γ(C_{1}^{ν}, O_{C_{1}^{ν}} (6p))) = ℙ^{5} given by |O_{C_{1}^{ν}} (6p)|. The projection is from a point on the tangent line T_{p}(C_{1}^{ν}), creating the cusp q.

Consider the normal form E := {x_{0}^{2}x_{2} = x_{1}(x_{1} − x_{2})(x_{1} − ℓx_{2})} ⊂ ℙ^{2} of C_{1}^{ν} given by |O_{C_{1}^{ν}} (3p)| where p = [1, 0, 0]. Then η(C_{1}^{ν}) ⊂ ℙ^{5} is the image of E under the second Veronese embedding

(⋆)  \begin{matrix}v_{2} : ℙ^{2} & −) & ℙ^{5} \\ [x_{0}, x_{1}, x_{2}] & ) & [x_{0}^{2}, x_{0}x_{1}, x_{1}^{2}, x_{0}x_{2}, x_{1}x_{2}, x_{2}^{2}]. \end{matrix}

The tangent line T to E at p is {x_{2} = 0}. If y_{0}, …, y_{5} are the homogeneous coordinates of ℙ^{5} in (⋆), the second Veronese image of T is given by the ideal ⟨y_{3}, y_{4}, y_{5}, y_{1}^{2} − y_{0}y_{2}⟩. Hence the tangent line to v_{2}(T) at p = η([1, 0, 0]) = [1, 0, …, 0] is {y_{0} = y_{3} = y_{4} = y_{5} = 0} and y_{1} is a local parameter of v_{2}(T) at p. Since v_{2}(T) and η(C_{1}^{ν}) agree to order one, it follows that the tangent line to η(C_{1}^{ν}) at p = η([1, 0, 0]) = [1, 0, …, 0] is {y_{0} = y_{3} = y_{4} = y_{5} = 0} and y_{1} is a local parameter of O_{η(C_{1}^{ν} ),p}. Therefore, the projection

\begin{matrix}pr : ℙ^{5} & −−→ & ℙ^{4} \\ [y_{0}, …, y_{5}] & ) & [y_{0}, y_{2}, y_{3}, y_{4}, y_{5}] \end{matrix}

kills the tangent direction and replaces p with a cusp.

The ideal I_{v_{2}(E)} of η(C_{1}^{ν}) = v_{2}(E) ⊂ ℙ^{5} is generated by:

y_{1}^{2} − y_{0}y_{2}, y_{4}^{2} − y_{2}y_{5}, y_{3}y_{4} − y_{1}y_{5}, y_{3}^{2} − y_{0}y_{5}, y_{2}y_{3} − y_{1}y_{4}, y_{1}y_{3} − y_{0}y_{4}, y_{2}y_{4}ℓ − y_{4}y_{5}ℓ − y_{2}^{2} + y_{0}y_{4} + y_{0}y_{5} + y_{2}y_{5}, y_{2}y_{5}ℓ − y_{4}y_{5}ℓ − y_{2}y_{4} + y_{0}y_{5} + y_{2}y_{5},

y_{1}y_{4}ℓ − y_{1}y_{5}ℓ − y_{1}y_{2} + y_{0}y_{3} + y_{1}y_{4} The ideal of pr ∘ η(C_{1}^{ν}) ⊂ ℙ^{4} is the kernel of the homomorphism

\begin{matrix}k[z_{0}, …, z_{4}] & ) k[y_{0}, …, y_{5}]/I_{v_{2}(E)} \\ (z_{0}, …, z_{4}) & ) (y_{0}, y_{2}, y_{3}, y_{4}, y_{5}). \end{matrix}

It is generated by:

\begin{matrix}z_{3}^{2} − z_{1}z_{4}, z_{2}^{2} − z_{0}z_{4}, z_{1}z_{4}ℓ − z_{3}z_{4}ℓ − z_{1}z_{3} + z_{0}z_{4} + z_{1}z_{4}, \\ z_{1}z_{3}ℓ − z_{3}z_{4}ℓ − z_{1}^{2} + z_{0}z_{3} + z_{0}z_{4} + z_{1}z_{4} \end{matrix}

(B) Second, we compute the Gröbner basis of I_{C_{1}} with respect to the total weight order:

(3.2)  \begin{matrix}z_{1}z_{4} − z_{3}^{2}, z_{1}z_{3} − z_{2}^{2} − z_{3}^{2}ℓ − z_{3}^{2} + z_{3}z_{4}ℓ, z_{0}z_{4} − z_{2}^{2}, \\ z_{0}z_{3} − z_{1}^{2} + z_{2}^{2}ℓ + z_{2}^{2} + z_{3}^{2}ℓ^{2} + z_{3}^{2}ℓ + z_{3}^{2} − z_{3}z_{4}ℓ^{2} − z_{3}z_{4}ℓ, \\ z_{2}^{2}z_{4} − z_{3}^{3} + z_{3}^{2}z_{4}ℓ + z_{3}^{2}z_{4} − z_{3}z_{4}^{2}ℓ, \\ z_{0}z_{2}^{2} − z_{1}^{3} + 2z_{1}z_{2}^{2}ℓ + 2z_{1}z_{2}^{2} + z_{2}^{2}z_{3}ℓ^{2} + z_{2}^{2}z_{3} \\ +z_{3}^{3}ℓ^{3} + z_{3}^{3}ℓ^{2} + z_{3}^{3}ℓ + z_{3}^{3} − z_{3}^{2}z_{4}ℓ^{3} − z_{3}^{2}z_{4}ℓ^{2} − z_{3}^{2}z_{4}ℓ \end{matrix}

(C) From (B) we obtain a Gröbner basis for Ĩ_{C_{1}} with terms without α underlined:

z_{3}z_{4}ℓα^{4} − z_{3}^{2}ℓα^{2} − z_{3}^{2}α^{2}\underline{−z_{2}^{2} + z_{1}z_{3}}, \underline{−z_{2}^{2} + z_{0}z_{4}}, \underline{−z_{3}^{2} + z_{1}z_{4}},
−z_{3}z_{4}^{2}ℓα^{4} + z_{3}^{2}z_{4}ℓα^{2} + z_{3}^{2}z_{4}α^{2}\underline{−z_{3}^{3} + z_{2}^{2}z_{4}},

−z_{3}z_{4}ℓ^{2}α^{6} − z_{3}z_{4}ℓα^{6} + z_{3}^{2}ℓ^{2}α^{4} + z_{3}^{2}ℓα^{4} + z_{3}^{2}α^{4} + z_{2}^{2}ℓα^{2} + z_{2}^{2}α^{2}\underline{−z_{1}^{2} + z_{0}z_{3}},

−z_{3}^{2}z_{4}ℓ^{3}α^{8} − z_{3}^{2}z_{4}ℓ^{2}α^{8} + z_{3}^{3}ℓ^{3}α^{6} − z_{3}^{2}z_{4}ℓα^{8} + z_{3}^{3}ℓ^{2}α^{6} + z_{3}^{3}ℓα^{6}
+z_{2}^{2}z_{3}ℓ^{2}α^{4} + z_{3}^{3}α^{6} + z_{2}^{2}z_{3}α^{4} + 2z_{1}z_{2}^{2}ℓα^{2} + 2z_{1}z_{2}^{2}α^{2}\underline{−z_{1}^{3} + z_{0}z_{2}^{2}}

(D) Substituting α = 0, we obtain the ideal of the flat limit:

⟨z_{3}^{2} − z_{1}z_{4}, z_{1}z_{3} − z_{0}z_{4}, z_{2}^{2} − z_{0}z_{4}, z_{1}^{2} − z_{0}z_{3}⟩

This is precisely the ideal of the tri-canonical model of C_{0}, regardless of ℓ.

3.3. Hilbert unstable curves - Instability of elliptic bridges.

Definition 5. An elliptic tail (resp. elliptic bridge) is a connected subcurve of arithmetic genus one meeting the rest of the curve in one node (resp. two nodes).

Figure 2. Generic elliptic bridges

Figure 3. Flat limit of ρ(t).C

In this section, we shall prove that a bicanonically embedded elliptic bridge is Hilbert unstable. Readers looking for context as to why this particular stability problem is important are invited to take a look at [HH07] and [HL09].

Let C be a generic elliptic bridge of genus g consisting of a genus g − 2 curve D meeting in two nodes q and r with a genus one subcurve E.

Proposition 5. Let C_{0} be the curve in Figure 3 consisting of D and two conics C_{1} and C_{2}, where D is embedded by |ω_{D}^{⊗2}(2q+2r)| and C_{1} and C_{2} meet D in nodes q and r respectively and meet each other in a tacnode. Then there is a one-parameter subgroup ρ : 𝔾_{m} ) SL_{N+1} such that

(1) ρ(t) · C specializes to a bicanonical c-semistable curve C_{0};

(2) P_{C_{0},ρ}(m) = −3(g − 1)(m − 1). In particular, C_{0} is Hilbert unstable. Since P_{C,ρ}(m) = P_{C_{0},ρ}(m), it follows that C is also Hilbert unstable.

20 BRENDAN HASSETT, DONGHOON HYEON AND YONGNAM LEE

Proof. Note that ω_{C}^{⊗2}|_{D} = ω_{D}^{⊗2}(2q + 2r) and ω_{C}^{⊗2}|_{E} = O_{E}(2q + 2r), which imply that D and E are embedded in linear subspaces of ℙ^{3g−4} of dimensions 3g − 6 and 3, respectively. Hence we can choose coordinates such that

\begin{matrix}x_{N−1} = x_{N} = 0 on D \\ x_{0} = ⋯ = x_{N−4} = 0 on E. \end{matrix}

We can extract equations for E embedded by |2q + 2r| by argument similar to extracting the normal form of elliptic curve embedded in ℙ^{2} by |3p_{0}|.

Let {1_{q}, x} and {1_{r}, y} be bases for Γ(2q) and Γ(2r), respectively. We may assume that 2q ≢ 2r: we first prove that an elliptic bridge with 2q ≢ 2r is unstable, and can deduce that an elliptic bridge with 2q ≡ 2r is also unstable since the unstable locus is closed. Under this assumption, we can choose x and y such that x ∈ Γ(2q − r) and y ∈ Γ(2r − q) and hence the vanishing order at q and r (on E) are as follows:

1_{q} x 1_{r} y
ord_{q} 2 0 0 1
ord_{r} 0 1 2 0

(3.3)  \begin{aligned}& x · 1_{r} & y · 1_{q} & xy & 1_{q} · 1_{r} \\ ord_{q} & 0 & 3 & 1 & 2 \\ ord_{r} & 3 & 0 & 1 & 2 \end{aligned}

Let x_{N−3} = x·1_{r}, x_{N−2} = y·1_{q}, x_{N−1} = xy and x_{N} = 1_{q}·1_{r}. One sees immediately that the image of E under |2q + 2r| lies on the Segre surface

{f_{1} := x_{N−3}x_{N−2} − x_{N−1}x_{N} = 0}.

Also, since dimΓ(4q + 4r) = 8, there is a nontrivial linear relation between the 9 elements

\begin{matrix}1 & x & y & xy & x^{2} & y^{2} & x^{2}y & xy^{2} & x^{2}y^{2} \\ x_{N}^{2} & x_{N−3}x_{N} & x_{N−2}x_{N} & x_{N−3}x_{N−2} & x_{N−3}^{2} & x_{N−2}^{2} & x_{N−3}x_{N−1} & x_{N−2}x_{N−1} & x_{N−1}^{2} \end{matrix}

Let f_{2} denote a linear relation:

\begin{matrix}f_{2} := c_{0}x_{N−3}^{2} + c_{1}x_{N−3}x_{N−1} + c_{2}x_{N−3}x_{N} + c_{3}x_{N−2}^{2} + c_{4}x_{N−2}x_{N−1} \\ +c_{5}x_{N−2}x_{N} + c_{6}x_{N−1}^{2} + c_{7}x_{N−1}x_{N} + c_{8}x_{N}^{2}. \end{matrix}

Because of our choice of coordinates that have specific vanishing orders at q and r, it follows that

\begin{matrix}(A) T_{q}E = \{x_{0} = ⋯ = x_{N−4} = x_{N−2} = x_{N} = 0\}, q = [0, …, 0, 1, 0, 0, 0] \\ (B) T_{r}E = \{x_{0} = ⋯ = x_{N−4} = x_{N−3} = x_{N} = 0\}, r = [0, …, 0, 0, 1, 0, 0]. \end{matrix}

(A) implies that c_{0} = c_{1} = 0 and c_{2} ≠ 0 while (B) forces c_{3} = c_{4} = 0 and c_{5} ≠ 0. Moreover, for E to be smooth, c_{6} must not be zero.

Taking all these into account, the generic form of f_{2} is as follows.

f_{2} = x_{N−1}^{2} + x_{N−3}x_{N} + x_{N−2}x_{N} + c_{1}x_{N−1}x_{N} + c_{2}x_{N}^{2}, c_{i} ∈ k.

The j-invariant of E can be computed by realizing it as a double cover of ℙ^{1} ([HL07a]) via E ) ℙ^{1} × ℙ^{1} )^{π} ℙ^{1} where π is the projection to one of the factors:

j(E) = \frac{−2^{8}3^{3}(c_{1}^{2} − 12c_{2})^{3}}{4(c_{1}^{2} − 12c_{2})^{3} + 27(2c_{1}^{3} − 72c_{1}c_{2} − 2^{4}3^{3})} .

Let ρ : 𝔾_{m} ) GL_{N+1}(k) be a one-parameter subgroup defined by the diagonal matrix

(3.4)  ρ(t) = ( \begin{matrix}t^{2} \\ & . & . & . \\ & & & & t^{2} \\ & & & & & t \\ & & & & & & 1 \end{matrix} )

To compute μ([C]_{m}, ρ), we shall first compute the ideal of the flat limit C_{0} of the family ρ(t) · C.

For fixed t ≠ 0, the two generators f_{1} and f_{2} of the ideal of C give rise to

f̃_{1}(x_{0}, …, x_{N}) = t^{4}f_{1}(t^{−2}x_{0}, t^{−2}x_{1}, …, t^{−2}x_{N−2}, t^{−1}x_{N−1}, x_{N})
= x_{N−3}x_{N−2} − t^{3}x_{N−1}x_{N} ⇝ x_{N−3}x_{N−2} = in_{≺_{ρ}}(f_{1})
f̃_{2}(x_{0}, …, x_{N}) = t^{2}f_{1}(t^{−2}x_{0}, t^{−2}x_{1}, …, t^{−2}x_{N−2}, t^{−1}x_{N−1}, x_{N})
= x_{N−1}^{2} + x_{N−3}x_{N} + x_{N−2}x_{N} + tc_{1}x_{N−1}x_{N} + t^{2}c_{2}x_{N}^{2}
⇝ x_{N−1}^{2} + x_{N−3}x_{N} + x_{N−2}x_{N} = in_{≺_{ρ}}(f_{2})

Let I′ = ⟨x_{N−3}x_{N−2}, x_{N−1}^{2} + x_{N−3}x_{N} + x_{N−2}x_{N}⟩ ⊃ in_{≺_{ρ}}(I_{E}) where I_{E} = ⟨f_{1}, f_{2}⟩ is the homogeneous ideal of E. The Hilbert polynomial of Proj(S/I′) is P(m) = 4m which is the same as the Hilbert polynomial m · deg O_{E}(2q + 2r) + 1 − 1 of the flat limit. Since I′ ⊃ in_{≺_{ρ}}(I_{E}), we conclude that I′ is equal to in_{≺_{ρ}}(I_{E}), the ideal defining the flat limit.

The curve E′ of arithmetic genus 1 defined by I′ consists of two conics

(3.5)  \begin{matrix}C′_{1} = \{x_{N−3} = 0, x_{N−1}^{2} + x_{N−2}x_{N} = 0\} \\ C′_{2} = \{x_{N−2} = 0, x_{N−1}^{2} + x_{N−3}x_{N} = 0\} \end{matrix}

meeting in a tacnode p′ = [0, …, 0, 1]. The flat limit of ρ(t) · D is obviously D itself since ρ acts trivially on D.

It remains to show that [C]_{m} is strictly semistable with respect to ρ. Equivalently, we may show that μ([C_{0}]_{m}, ρ) = 0. Let I_{0} denote the ideal of C_{0}. Recall that μ([C]_{m}, ρ) = μ([C_{0}]_{m}, ρ), the right hand side of which we shall compute by using the formula

−(N + 1) ∑_{i=1}^{P(m)} wt_{ρ}(x^{a(i)} ) + mP(m) ∑_{i=0}^{N} r_{i}

where x^{a(1)}, …, x^{a(P(m))} are degree m monomials not in in_{≺}(I_{0}), and r_{i} = wt_{ρ}(x_{i}).

First, we shall consider the second Hilbert point of C_{0}. We need to sort out the degree 2 monomials (of weight < 4) that are not in in_{≺}(I′). The following are the degree 2 monomials with ρ-weight less than 4:

ρ-weight
3  x_{j}x_{N−1}, j ≤ N − 2
2  x_{N−1}^{2}, x_{j}x_{N}, j ≤ N − 2
1  x_{N−1}x_{N}
0  x_{N}^{2}

Among these, clearly x_{j}x_{N} and x_{j}x_{N−1}, j = 0, …, N−4, are of weight < 4 and in in_{≺}(I_{0}) since they are in I_{0}. Therefore, the only degree 2 monomials of weight < 4 that are possibly not in in_{≺}(I_{0}) are

(3.6)  \begin{matrix}ρ-weight \\ 3 & x_{N−3}x_{N−1}, x_{N−2}x_{N−1} \\ 2 & x_{N−3}x_{N}, x_{N−2}x_{N}, x_{N−1}^{2} \\ 1 & x_{N−1}x_{N} \\ 0 & x_{N}^{2} \end{matrix}

I claim that in the table (3.6), x_{N−3}x_{N} is the only monomial that is in in_{≺}(I_{0}). Clearly, in_{≺}(I_{0}) ⊂ in_{≺}(I′). A Gröbner basis of I′ is:

x_{N−3}x_{N} + x_{N−2}x_{N} + x_{N−1}^{2}, x_{N−3}x_{N−2}, x_{N−2}^{2}x_{N} + x_{N−2}x_{N−1}^{2}.

Hence the initial ideal is

(3.7)  in_{≺}(I′) = ⟨x_{N−3}x_{N}, x_{N−3}x_{N−2}, x_{N−2}^{2}x_{N}⟩

and the only degree 2 monomials in in_{≺}(I′) are x_{N−3}x_{N} and x_{N−3}x_{N−2}. Hence among the monomials in the list, x_{N−3}x_{N} is the only possible element in in_{≺}(I_{0}).

Since f_{2} = x_{N−1}^{2} + (x_{N−3} + x_{N−2})x_{N} ∈ I′ vanishes entirely on D, f_{2} ∈ I_{0} and

in_{≺}(f_{2}) = x_{N−3}x_{N} ∈ in_{≺}(I_{0}).

On the other hand, if x_{N−3}x_{N−2} = in_{≺}(f) for some f ∈ I_{0}, then f must be of the form

ax_{N−3}x_{N−2} + bx_{N−2}^{2} + x_{N−1}g_{1} + x_{N}g_{2},

for some a, b ∈ k and linear polynomials g_{1}, g_{2}. But this would imply that x_{N−2} = 0 or ax_{N−3} + bx_{N−2} = 0 entirely on D, which contradicts that D is nondegenerate in {x_{N−1} = x_{N} = 0}. Hence x_{N−3}x_{N−2} ∉ in_{≺}(I_{0}). Therefore, the total weight ∑ _{i=1}^{P(2)} wt_{ρ}x^{a(i)} is

\begin{aligned}∑ _{i=1}^{P(2)} wt_{ρ}(x^{a(i)} ) & = 2 + 2 · 3 + 2 + 1 + 4 · (7g − 7 − 6) \\ & = 11 + 28g − 52 = 28g − 41. \end{aligned}

On the other hand,

\begin{aligned}\frac{2 · P(2)}{3g − 3} ∑_{i=0}^{N} r_{i} & = \frac{2 · 7(g − 1)}{3(g − 1)} · (1 + 2(3g − 5)) \\ & = 28g − 42. \end{aligned}

Hence

μ([C]_{2}, ρ) = μ([C_{0}]_{2}, ρ)
= −(3g − 3) · (28g − 41 − (28g − 42))
= −3(g − 1) < 0.

It follows that ρ destabilizes the 2nd Hilbert point of C.

Now let’s consider the 3rd Hilbert point of C. The degree 3 monomials of ρ-weight less than 6 are

ρ-weight
5  x_{i}x_{j}x_{N−1}, i, j ≤ N − 2
4  x_{j}x_{N−1}^{2}, x_{i}x_{j}x_{N}, i, j ≤ N − 2
3  x_{N−1}^{3}, x_{i}x_{N−1}x_{N}, i ≤ N − 2
2  x_{N−1}^{2}x_{N}, x_{j}x_{N}^{2}, j ≤ N − 2
1  x_{N−1}x_{N}^{2}
0  x_{N}^{3}

Among these monomials, x_{i}x_{j}x_{N−1}, x_{j}x_{N−1}^{2}, x_{i}x_{j}x_{N} and x_{j}x_{N}^{2}, are obviously contained in in_{≺}(I_{0}) if i ≤ N − 4 or j ≤ N − 4 since they are in I_{0}. Hence we need to consider

(3.8)  \begin{matrix}ρ-weight \\ 5 & x_{N−3}^{2}x_{N−1}, x_{N−3}x_{N−2}x_{N−1}, x_{N−2}^{2}x_{N−1} \\ 4 & x_{N−3}^{2}x_{N}, x_{N−3}x_{N−2}x_{N}, x_{N−2}^{2}x_{N}, x_{N−3}x_{N−1}^{2}, x_{N−2}x_{N−1}^{2} \\ 3 & x_{N−1}^{3}, x_{N−3}x_{N−1}x_{N}, x_{N−2}x_{N−1}x_{N} \\ 2 & x_{N−1}^{2}x_{N}, x_{N−3}x_{N}^{2}, x_{N−2}x_{N}^{2} \\ 1 & x_{N−1}x_{N}^{2} \\ 0 & x_{N}^{3} \end{matrix}

First, note that x_{N−3}x_{N−2}x_{N−1} and x_{N−3}x_{N−2}x_{N} are in in_{≺}(I_{0}) since they are in I_{0}. Then we argue similarly as in the 2nd Hilbert point case. By examining the initial ideal (3.7), we deduce that among the monomials in (3.8), the following monomials are the only possible elements in in_{≺}(I_{0}):

x_{N−3}x_{N−2}x_{N−1}, x_{N−3}^{2}x_{N}, x_{N−3}x_{N−2}x_{N}, x_{N−3}x_{N−2}x_{N−1}, x_{N−2}x_{N−1}^{2}, x_{N−3}x_{N}^{2}.

We have

\begin{aligned}g_{1} & = x_{N−2} · (x_{N−1}^{2} + (x_{N−3} + x_{N−2})x_{N}) − x_{N} · (x_{N} − 3x_{N−2}) \\ & = x_{N−2}^{2}x_{N} + x_{N−2}x_{N−1}^{2} ∈ I_{0} \end{aligned}

Hence x_{N−2}^{2}x_{N} = in_{≺}(g_{1}) ∈ in_{≺}(I_{0}). Therefore, the total weight is

\begin{aligned}∑ _{i=1}^{P(3)} wt_{ρ}(x^{a(i)} ) & = 2 · 5 + 2 · 4 + 2 · 3 + 2 · 2 + 1 · 1 + 6 · (11(g − 1) − 10) \\ & = 29 + 6(11g − 21) = 66g − 97. \end{aligned}

On the other hand,

\begin{aligned}\frac{3 · P(3)}{3g − 3} ∑_{i=0}^{N} r_{i} & = \frac{3 · 11(g − 1)}{3(g − 1)} · (1 + 2(3g − 5)) \\ & = 66g − 99. \end{aligned}

Hence

μ([C]_{3}, ρ) = μ([C_{0}]_{3}, ρ)
= −(3g − 3) · (66g − 97 − (66g − 99))
= −6(g − 1) < 0.

It follows that ρ destabilizes the 3rd Hilbert point of C. From μ([C_{0}]_{2}, ρ) and μ([C_{0}]_{3}, ρ), we obtain the filtered Hilbert function

\begin{aligned}P_{C_{0},ρ}(m) & = (m − 1)[−3(g − 1)(3 − m) − 6(g − 1)(m/2 − 1)] \\ & = −3(g − 1)(m − 1). \end{aligned}

This has negative values for all m ≥ 2, and it follows that [C]_{m} is unstable with respect to ρ for all m ≥ 2.

□

Corollary 6. Let C_{0} and ρ be as in Proposition 5. Then μ(Ch(C_{0}), ρ) = 0.

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(BH) Department of Mathematics, Rice University, MS 136, Houston, TX 77251-1892, U.S.A., email: hassett@rice.edu

(DH) Department of Mathematics, Marshall University, Huntington, WV 25755, U.S.A., email: hyeond@marshall.edu

(YL) Department of Mathematics, Sogang Univeristy, Sinsu-dong, Mapo-gu, Seoul 121-742, Korea, email: ynlee@sogang.ac.kr