Stability of pencils of plane sextics and Halphen pencils of index two

We study the problem of classifying pencils of plane sextics up to projective equivalence via geometric invariant theory (GIT). In particular, we provide a complete description of the GIT stability of certain pencils of plane sextics called Halphen pencils of index two-classical geometric objects which were first introduced by G. Halphen in 1882. Inspired by the work of Miranda on pencils of plane cubics, we obtain explicit stability criteria in terms of the types of singular fibers appearing in their associated rational elliptic surfaces.


Introduction
In this paper we study the stability of pencils of plane curves of degree six under projective equivalence in the sense of geometric invariant theory (GIT).Building in the results we obtained in [16] and [17], we completely describe the stability of Halphen pencils of index two -classical geometric objects first introduced by Halphen in 1882 [5].These are pencils of plane curves of degree six with exactly nine base points (possibly infinitely near) of multiplicity two (Definition 4.3).
More generally, a Halphen pencil (of index m) corresponds to a rational surface Y that admits a fibration f : Y → P 1 with generic fiber a smooth curve of genus one (see e.g.[2, Chapter V, §6]).These surfaces are called rational elliptic surfaces (of index m), and the correspondence is given by the blow-up of P 2 at the nine base points of the pencil.They have exactly one multiple fiber of multiplicity m, and f does not have a section if and only if m > 1.In this paper we are interested in the case m = 2.
We follow an approach similar to that of Miranda in [11], and we provide a complete and geometric characterization of the stability of Halphen pencils of index two in terms of the types of singular fibers appearing in the associated rational elliptic surfaces (of index two).In [11], Miranda describes the stability of pencils of plane cubics up to projective equivalence, which leads to a compactification of the moduli space of rational elliptic surfaces with section.Such compactification agrees with the one obtained by the same author in [12], where the surfaces are described by their Weierstrass models.When the existence of a global section is not assumed, there is no analogue for the Weierstrass model and even the dimensions of the parameter spaces involved are much higher, which makes describing the stability conditions a much harder problem.The results we obtained in [16] and [17] have helped us to overcome such difficulties.
The main results of this paper are given by Theorems 1.1, 1.2 and 1.3 below, where P denotes a Halphen pencil of index two and Y denotes the associated rational elliptic surface.We write the pencil P as λB + µ(2C) = 0, where C is the unique cubic through the nine base points of P and the curve B corresponds to some (non-multiple) fiber, which we denote by F .Theorem 1.1.When C is smooth the pencil P is stable if and only if one of the following statements hold (i) all fibers of Y are reduced (ii) Y contains at most one non-reduced fiber F of type I * n or IV * (iii) there exists exactly one (non-multiple) fiber F in Y such lct(Y, F ) ≤ 1/4 and B is semistable (iv) Y contains two fibers of type I * 0 and there is no one-parameter subgroup λ that destabilizes the two corresponding curves simultaneously.
Theorem 1.2.When C is singular the pencil P is stable if and only if one of the following statements hold (i) all fibers of Y are reduced (ii) P contains at worst two strictly semistable curves and there is no one-parameter subgroup λ that destabilizes these two curves simultaneously (iii) Y contains a fiber of type IV * and B is unstable Theorem 1.3.The pencil P is semistable if and only if every curve in P is semistable or Y does not contain a fiber F of type II * .
We observe that every pencil of plane curves can also be seen as a holomorphic foliation of P 2 , in particular any Halphen pencil of index two.In [10], the authors give a criterion for when a foliation of P 2 of degree seven is a Halphen pencil of index two.In [1], a stratification of the space of foliations on P 2 of degree d is given (in particular d = 7), thus providing an alternative description of the semistable and unstable Halphen pencils of index two under projective equivalence -viewed as foliations.However, such stratification does not provide a description of the stable pencils, which we do in this paper.
Organization.The paper is organized as follows: In Section 2 we describe the vanishing criteria in terms of Plücker coordinates for (semi)stability of pencils of plane sextics.Then, a more geometric description is presented in Section 3, where we translate these vanishing criteria into equations for the generators of the pencil.
Finally, in Section 4 we provide a complete characterization of the (semi)stability of Halphen pencils of index two.

Stability Criterion for Pencils of Plane Sextics
Following the ideas in [11], we view a pencil of plane sextics as a choice of a line in the space of all plane curves of degree six.In other words, we identify the space P 6 of all such pencils with the Grassmannian Gr(2, S 6 V * ), where V .= H 0 (P 2 , O P 2 (1)).The latter, in turn, can be embedded in P(Λ 2 S 6 V * ) via Plücker coordinates so that a pencil corresponds to a decomposable 2−form.The automorphism group of P 2 , P GL(3), acts naturally on V , hence on the invariant subvariety P 6 .
Our main tool for determining which are the stable and strictly semistable pencils is the numerical criterion of Hilbert-Mumford.Recall that the Hilbert-Mumford criterion says a pencil P ∈ P 6 is unstable (resp.not stable) if and only if there exists a 1-parameter subgroup C × → SL(V ) with respect to which all weights are positive (resp.non-negative).Thus, we need to obtain explicit vanishing conditions on the Plücker coordinates and for that we need to know how the diagonal elements in P GL(V ) act on these coordinates.In fact since P GL(V ) and SL(V ) act with the same orbits it is convenient to focus on the diagonal action of the latter.
If we choose a pencil P ∈ P 6 and two curves C f and C g as generators, these represented by f = f ij x i y j z 6−i−j and g = g ij x i y j z 6−i−j (respectively), then the Plücker embedding takes the 2 × 27 matrix f ij g ij to the point in P ( 28 2 )−1 whose coordinates are given by all the 2 × 2 minors Now, if and element of SL(V ) is given by in some choice of coordinates [x, y, z] in P 2 , then its action on the coordinates of P 2 is given by [x, y, z] → [αx, βy, γz].Thus, the action on a point (f ij ) ∈ S 6 V * representing f ij x i y j z 6−i−j is given by (f ij ) → (α i β j γ 6−i−j f ij ) and, therefore, it acts on the Plücker coordinates by So we can now express the Hilbert-Mumford criterion for a pencil P ∈ P 6 as the vanishing of some of its Plücker coordinates (m ijkl ) with respect to a convenient choice of basis.We will show that there are only (explicit) finitely many critical one-parameter subgroups λ that need to be tested when analyzing the positivity of the Hilbert-Mumford weight µ(P, λ) (defined in (1) below).
We may assume any one-parameter subgroup λ is normalized, meaning we choose coordinates [x, y, z] in P 2 so that we have for some weights a x , a y , a z ∈ Z with a x ≥ a y ≥ a z , a x > 0 and a x + a y + a z = 0.
In particular, the action of λ(t) in the Plücker coordinates is given by where e ijkl .= a x (2i + 2k + j + l − 12) + a y (2j + 2l + i + k − 12).Further, we often normalize the weights so that a x = 1, a y = a and a The sign of the function µ(P, λ) does not change under these reductions and the Hilbert-Mumford criterion becomes: Proposition 2.1.A pencil P ∈ P 6 is unstable (resp.not stable) if and only if there exists a rational number a ∈ [−1/2, 1] and coordinates in P 2 such that if the pencil is represented in those coordinates by (m ijkl ), then m ijkl = 0 whenever e ijkl (a) ≤ 0 (resp.e ijkl (a) < 0).
A priori, for each choice of coordinates in P 2 one would need to test all possible values of a ∈ [−1/2, 1] ∩ Q to verify the stability criterion.Because the function (for a fixed P and a choice of coordinates) (1) µ(P, λ) .= min{e ijkl (a) : is piecewise linear, a key observation is that we only need to test its positivity for a finite number of critical values a ∈ [−1/2, 1] ∩ Q.
In other words, the conditions e ijkl (a) ≤ 0 (resp.e ijkl (a) < 0) divide the interval [−1/2, 1] into finitely many subintervals [a n , a n+1 ] within which the truthfulness of the inequality remains constant.That is, for each interval [a n , a n+1 ] we can find values of i, j, k and l for which the inequality e ijkl (a) ≤ 0 (resp.e ijkl (a) < 0) remains true for all a ∈ [a n , a n+1 ].
The intervals we find are given by Lemmas 2.2 and 2.3 below: Lemma 2.2.The condition e ijkl (a) ≤ 0 divides the interval [−1/2, 1] into finitely many subintervals and in order to obtain minimal conditions for unstability, it suffices considering only the following six distinct subintervals: In particular, we can restate the criteria for unstability ( resp.non-stability) as in Theorem 2.4 (resp.Theorem 2.5): Theorem 2.4.A pencil P ∈ P 6 is unstable if and only if there exist coordinates in P 2 so that if the pencil is represented in those coordinates by (m ijkl ), then m ijkl = 0 whenever the (appropriate) values of i, j, k and l satisfy either one of the following conditions: (1) (2i + 2k + j + l − 12) − 13/42(2j Theorem 2.5.A pencil P ∈ P 6 is not stable if and only if there exist coordinates in P 2 so that if the pencil is represented in those coordinates by (m ijkl ), then m ijkl = 0 whenever the (appropriate) values of i, j, k and l satisfy either one of the following conditions: (1) Now, in order to know what is the set of values i, j, k and l for which the Plücker coordinates m ijkl vanish in Theorems 2.4 and 2.5 above, it is convenient to express these values in terms of the pairs (r, s).

A Geometric Description
In the previous section we have completely characterized the stability of a pencil P ∈ P 6 in terms of its Plücker coordinates (m ijkl ).We now want to understand which are the geometric properties unstable and non-stable pencils have.More precisely, we want to translate the stability criteria into equations for the generators of the pencil.
Throughout this section, given an unstable (resp.not stable) pencil P ∈ P 6 we choose coordinates [x, y, z] in P 2 as in Theorem 2.4 (resp.2.5) and generators C f and C g having defining polynomials (in these coordinates) f = f ij x i y j z 6−i−j and g = g ij x i y j z 6−i−j .Then, the idea is that each vanishing condition m ijkl = 0 translates into the vanishing of the coefficients of some pair C f ′ and C g ′ of generators (not necessarily the original pair).
To illustrate what kind of computations are involved in this process we prove Theorem 3.1 below.We use the notation m 1 , . . ., m n to denote the subspace of homogeneous polynomials of degree six in the variables x, y and z which is generated by the monomials m i .Whereas m 1 , . . ., m n denotes the subspace of those polynomials which are generated by all the monomials which are different from the m i .Theorem 3.1.A pencil P ∈ P 6 satisfies the vanishing conditions in case 1 of Theorem 2.7 if and only if there exist coordinates in P 2 and generators C f and C g of P such that either Case 1 f ∈ x 4 z 2 , x 4 yz, x 4 y 2 , x 5 z, x 5 y, x 6 and g is arbitrary Case 2 f ∈ x 3 z 3 , x 3 yz 3 , x 3 y 2 z, x 3 y 3 , x 4 z 2 , x 4 yz, x 4 y 2 , x 5 z, x 5 y, x 6  and g ∈ z 6 , yz 5 , y 2 z 4 , y 3 z 3 , y 4 z 2 , y 5 z, y 6 Case 3 f and g ∈ x i y j z 6−i−j , where 2 ≤ i ≤ 6, 0 ≤ j ≤ 6 and i + j ≤ 6 Proof.Let us assume P is not stable and that its Plücker coordinates (m ijkl ) must vanish for all i, j, k and l satisfying i + k = r and j + l = s for all the pairs (r, s) in case 1 of Theorem 2.7.Using the relations m ijkl = −m klij and m ijij = 0 we can compute the minimal set of values {i, j, k, l} (in order) so that the m ijkl vanish.
Note that we have introduced an ordering on the Plücker coordinates coming from the restrictions on i, j, k and l.So, the first step is to look at the equation m ijkl = 0 for the first term {i, j, k, l} in the list above, namely we look at the equation m 0001 = 0.It follows that either (1) f 00 = g 00 = 0 or (2) g 00 = 0 or (3) f 00 = 0 Moreover, if (2) (or (3) by symmetry) holds, then taking f ′ = f − f 00 g 00 g we can assume f 00 = 0 and we must have f 01 = 0.The next step then is, at each of the cases above, to look at the next vanishing condition m 0002 = 0 coming from the second term {i, j, k, l} in the list.Again there are three possibilities: Either f 00 = g 00 = 0 or g 02 = 0 or f 02 = 0.
We proceed in this manner until there are no more equations m ijkl = 0 to solve.In fact our list tells us that m 00kl vanish for all (appropriate) 0 ≤ k ≤ 3 and 0 ≤ l ≤ 6.Thus, our algorithm tells us that if we are in the situation of case (2), then one of the generators belongs to x k j l z 6−k−l for all kl such that m 00kl = 0. And, by symmetry, we reach the same conclusion if (3) holds.A similar reasoning applies to the next set of vanishing conditions m 01kl = 0 and so on.
It is important to note, however, that at each step, when solving the equations m ijkl = 0 we have to take into account whether there are or there are not previous conditions on the coefficients f ij , g ij , f kl and g kl .
Following the sketched algorithm we obtain the desired geometric description of the pencil P.
Note that the same algorithm outlined above in the proof of Theorem 3.1 can be applied whenever P is unstable (resp.not stable) and satisfies one of the vanishing conditions in anyone of the cases in Theorem 2.6 (resp.2.7).However, the computations involved are very lengthy and the assistance of a computer is needed.And even the corresponding statements as in Theorem 3.1 require several pages to be presented.
The complete geometric description of the stability conditions in terms of equations for the generators is available upon request.And instead of exhibiting these tiresome results we will present next (without proofs) only those results that are essential in the study of Halphen pencils of index two (Section 4) and we will mostly focus on proper pencils: Definition 3.2.A pencil P ∈ P 6 is called proper if any two curves on it intersect properly, meaning its base locus is zero dimensional, i.e. it consists of a finite number of points.

Stability of Halphen pencils of index two
We now combine the results from Sections 2 and 3, together with the results we have previously obtained in [16] and [17], in order to completely characterize the stability of Halphen pencils of index two.Similar to the work of Miranda we provide a geometric description in terms of the types of singular fibers of the corresponding rational elliptic surface.
The relevant definitions are as follows: Definition 4.1.A rational elliptic surface consists of a smooth and projective rational surface Y together with a fibration (a surjective proper flat morphism) f : Y → P 1 such that the generic fiber is a smooth genus one curve and there are no (−1)−curves in any fiber.Any rational elliptic surface has finitely many singular fibers, and these have been classified by Kodaira and Néron [7,8,13].Table 1 below gives the full classification.Over a field of characteristic zero, any multiple fiber is of type I n for some n ≥ 0 [2, Proposition 5.1.8] .The correspondence between Halphen pencils and rational elliptic surfaces is given by the following Proposition: ).Let f : Y → P 1 be a rational elliptic surface of index m and let F be a choice of a fiber of f , then there exists a birational map π : Y → P 2 so that f • π −1 is a Halphen pencil of index m and, moreover, B .= π(F ) is a plane curve of degree 3m.Conversely, given a Halphen pencil of index m, taking the minimal resolution of its base points we obtain a rational elliptic surface of index m.

Kodaira Type Number of Components Dual Graph
In particular, any Halphen pencil of index m contains exactly one multiple cubic mC, which corresponds to the unique multiple fiber in the associated rational elliptic surface [2, Proposition 5.61,(iii)].Thus any Halphen pencil P of index m can be written in the following form λ(B) + µ(mC) = 0, where the curve B corresponds to some (non-multiple) fiber of Y that we denote by F .
With these notations in mind we now focus in the case m = 2.In Section 4.1 we give some sufficient conditions for the (semi)stability of P and in Section 4.2 we describe some necessary conditions.A complete characterization is then obtained by further studying the stability of P when the fiber F is of type II * , III * or IV * (Sections 4.3, 4.4 and 4.5).
4.1.Sufficient conditions for (semi)stability.We first establish sufficient conditions for the (semi)stability of a Halphen pencil of index two.We will see that the log canonical thresholds (see e.g.[9, Section 8]) of the pairs (P 2 , 2C) and (P 2 , B) plays a fundamental role.Theorem 4.5.If P is not stable, then Y contains a non-reduced fiber.
Proof.Since lct(P Proof.From the classification in [14] we know that any other fiber of Y is reduced.By [16,Propositions 4.14 and 4.15] we also know that all other curves in P are reduced and have log canonical threshold greater than 1/2.As observed in [4] and [6], this implies all the curves in P are stable except for one curve that is semistable.
Corollary 4.8.2.If C is smooth and Y contains exactly one fiber F of type I * n , n ≤ 4, then P is stable.
Proof.Again, from the classification in [14] we know that any other fiber of Y is reduced.Since the curve B is such that lct(P 2 , B) ≥ 1/2, hence it is semistable [4,6], we can argue as in the proof of Corollary 4.8.1 to conclude all the curves in P are stable except (possibly) for one curve that is semistable.Theorem 4.9.If Y contains two fibers of type I * 0 , then P is strictly semistable if and only if there exists a one-parameter subgroup λ (and coordinates in P 2 ) such that the two curves corresponding to the fibers of type I * 0 are both non-stable with respect to this λ.
Proof.By [16,Proposition 4.16], if F is a fiber of type I * 0 , then the corresponding plane curve B is such that lct(P 2 , B) ≥ 1 2 , hence it is semistable [4,6].The result then follows from [17,Theorem 1.6].Note that from the topological Euler characteristic of Y we know C has to be smooth, hence stable [15].
And when C is singular we prove: Theorem 4.10.If C is singular and Y contains exactly one fiber F of type I * n , n ≤ 4, then P is strictly semistable if and only if there exists a one-parameter subgroup λ (and coordinates in P 2 ) such that 2C and B = π(F ) are both non-stable with respect to this λ.
Proof.Since both 2C and B are semistable and all other curves in P are stable, the result follows from [17,Theorem 1.6].
Theorem 4.11.If C is singular, Y contains a fiber F of type II * , III * and IV * and the curve B = π(F ) is semistable, then P is strictly semistable if and only if there exists a one-parameter subgroup λ (and coordinates in P 2 ) such that 2C and B are both non-stable with respect to this λ.
Proof.Again, the result follows from [17,Theorem 1.6] because both 2C and B are semistable and all other curves in P are stable.
Finally, in order to complete our description, we need to study the stability of P when F is a fiber of type II * , III * or IV * .4.3.The stability of P when F is of type II * .When F of type II * , then [16,Theorem 5.15] tells us B can only be realized by one of the following plane curves: (i) a triple conic (ii) a nodal cubic and an inflection line, with the line taken with multiplicity three (iii) two triples lines (iv) a conic and a tangent line, with the line taken with multiplicity four (v) a line with multiplicity five and another line If B is a triple conic, then B is strictly semistable [15].In this case, if C is smooth, then P is stable (Corollary 4.8.1) and if C is singular, then P is strictly semistable if and only if there exists a one-parameter subgroup λ (and coordinates in P 2 ) such that 2C and B are both non-stable with respect to this λ (Theorem 4.11).
When B is one of the curves in (ii), (iii), (iv) or (v) then we can use the explicit constructions obtained in [16] to conclude P is unstable.Proposition 4.12.If Y contains a fiber of type II * and P contains a curve consisting of a triple line and a nodal cubic, then P is unstable.
Proof.Let P and Y be as above.One can show that the line is an inflection line of both the nodal cubic and C, which is smooth [16,Example 7.57].
In particular, we can find coordinates in P 2 so that the curve B has equation x 3 (xz 2 − y 2 (y + x)) = 0 and C is given by x 2 y + xz 2 − y 3 − xy 2 = 0. Then the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (1) of Theorem 2.6 and we conclude P is unstable.Alternatively, we can easily check the equations for B and 2C belong to Case 4 of Theorem 3.5.Proposition 4.13.If Y contains a fiber of type II * and P contains a curve consisting of two triple lines, then P is unstable.
Proof.Let P and Y be as above.One can show that one of the lines is an inflection line of C and the other line must be tangent to the cubic with multiplicity two [16,Example 7.56].
In particular, we can find coordinates in P 2 so that B is given by x 3 y 3 = 0 and C is given by z 2 x − y(y − x)(y − α • x) = 0, where α ∈ C\{0, 1}.Then the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (1) of Theorem 2.6 and we conclude P is unstable.Alternatively, we can easily check the equations for B and 2C belong to Case 5 of Theorem 3.5.Proposition 4.14.If Y contains a fiber of type II * and P contains a curve consisting of a conic and a tangent line, with the line taken with multiplicity four, then P is unstable.
Proof.Let P and Y be as above.One can show that C must be tangent to the conic (resp.the line) at the point Q ∩ L with multiplicity six (resp.two) as in [16,Example 7.58].
In particular, we can find coordinates in P 2 so that B is given by the zeros of x 4 (y 2 + xz) and C is given by f = f ij x i y j z 6−i−j = 0, with f 00 = f 01 = f 02 = 0. Thus, the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (1) of Theorem 2.6 and we conclude P is unstable.Alternatively, we can easily check the equations for B and 2C belong to Case 2 of Theorem 3.5.Proposition 4.15.If Y contains a fiber of type II * and P contains a curve consisting of a line with multiplicity five and another line, then P is unstable.
Proof.Let B ∈ P be the curve consisting of a line with multiplicity five and another line.We can choose coordinates so that B is the curve x 5 (x − z) = 0 and C is the cubic y = π(F ) consists of a triple line and a nodal cubic, then P is not stable.
Proof.We can find coordinates in P 2 [16, Example 7.52] so that the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (4) of Theorem 2.7 and we conclude P is not stable.Alternatively, we can also apply Theorem 3.3.= π(F ) consists of a triple line, a conic and another line, then P is not stable.
Proof.Let P and Y be as above.We can find coordinates in P 2 as in [16,Example 7.51] so that the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (3) of Theorem 2.7 and we conclude P is not stable.Alternatively, we can also apply Theorem 3.3.So the remaining question is: Can P be unstable?We will show the answer to this questions is no.= π(F ) contains a line with multiplicity four, then P is semistable.
Proof.If P were unstable, then P (and B) would be as in (i) or (ii) in Lemma 4.27.In [16,Section 6] we show that this is not case for a fiber of type III * .Lemma 4.29.Let P be a Halphen pencil of index two containing a curve B such that B = 3L + C ′ , where L is a line and C ′ is a cubic (possibly reducible).Letting 2C denote the unique multiple cubic in P we have that if P is unstable, then either (1) L is an inflection line of C at a point where the intersection multiplicity of L and C ′ is ≥ 2.
(2) L is tangent to C at a point where the intersection multiplicity of L and C ′ is 3.
Proof.It follows from Theorem 3.7.= π(F ) contains a triple line, then P is semistable.
Proof.If P were unstable, then P (and B) would be as in (i) or (ii) in Lemma 4.29.In [16,Section 6] we show that this is not case for a fiber of type III * .4.5.The stability of P when F is of type IV * .Finally, we describe the stability of P when F is of type IV * .We will show that either P is stable or C is singular and B is semistable, in which case we are in the situation of Theorem 4.11.
Lemma 4.32.Let P be a Halphen pencil of index two containing a curve B such that B = 3L + C ′ , where L is a line and C ′ is a cubic (possibly reducible).Letting 2C denote the unique multiple cubic in P we have that if P is not stable, then either (i) L is an inflection line of C or (ii) L is tangent to C at a point where L and C ′ also intersect or (iii) there is a base point where L and C intersect and where the intersection multiplicity of L and C ′ is 3 Proof.It follows from Theorem 3.3.Proof.If P were not stable, then P (and B) would be as in (i),(ii) or (iii) in Lemma 4.32.In [16,Section 6] we show that this is not case for a fiber of type IV * .Lemma 4.34.Let P be a Halphen pencil of index two containing a curve B such that B = 2L + Q, where L is a line and Q is a quartic (possibly reducible).Letting 2C denote the unique multiple cubic in P we have that if P is not stable, then the intersection multiplicity of L and Q at some base point is 4.

Definition 4 . 2 .
Given f : Y → P 1 as above, we define the index of the fibration as the positive generator of the ideal{D • Y η ; D ∈ Pic(Y )}Z, where Y η is a generic fiber.

Definition 4 . 3 .
A Halphen pencil of index m is a pencil of plane curves of degree 3m with nine (possibly infinitely near) base points of multiplicity m.

Proposition 4 .
23.If Y contains a fiber F of type III * and B .

Proposition 4 .
24.If Y contains a fiber F of type III * and B .= π(F ) consists of two triple lines, then P is not stable.Proof.It follows from Lemma 4.18.Combining Propositions 4.17 through 4.24 and [16, Theorem 5.16] we obtain: Theorem 4.25.If Y contains a fiber F of type III * and B .= π(F ) is unstable, then P is not stable.

Remark 4 .
26.Note that when F is of type III * and B .= π(F ) is semistable we can refer to Corollary 4.8.1 and Theorem 4.11.

Lemma 4 . 6 .
27. Let P be a Halphen pencil of index two containing a curve B such that B = 4L + Q, where L is a line and Q is a conic (possibly reducible).Letting 2C denote the unique multiple cubic in P we have that if P is unstable, then either (i) L is an inflection line of C or (ii) L is tangent to C at a point where L and Q also intersect Proof.It follows from Theorem 3.Proposition 4.28.If Y contains a fiber F of type III * and B .

Proposition 4 .
30.If Y contains a fiber F of type III * and B .

Proposition 4 .
31.If Y contains a fiber F of type III * and B .= π(F ) consists of a double line, a cubic and another line, then P is semistable.Proof.It follows from Theorem 3.8.

Proposition 4 .
33.If Y contains a fiber F of type IV * and B .= π(F ) contains a triple line, then P is stable.
[15]proof is very similar to the proof of Theorem 4.5.Since we know Necessary conditions for (semi)stability.The next step is to obtain necessary conditions.Again, the log canonical thresholds of the pairs (P 2 , 2C) and (P 2 , B) will be fundamental to our analysis.WhenC is smooth and B is semistable we prove: It follows from[17, Theorem 1.5] and the fact that 2C is stable[15].Corollary 4.8.1.If C is smooth, F is of type II * , III * or IV * and B .= π(F ) is semistable, then P is stable.
rational elliptic surface Y → P 1 contains a non-reduced fiber F .Remark 4.6.A completely analogous argument in fact shows the statement of Theorem 4.5 is true for Halphen pencils of any index.Theorem 4.7.If P is unstable, then Y contains a fiber of type II * , III * or IV * .Proof.* , III * or IV * .4.2.Proposition 4.8.If C is smooth and all curves in P are stable except (possibly) for one curve that is semistable, then P is stable.Proof.
[16,16, Example 7.59].Then the Plücker coordinates of P satisfy the vanishing conditions of Case (1) in Theorem 2.6.Or, yet, we can easily check the equations for B and 2C belong to Case 1 of Theorem 3.5.Combining Propositions 4.12 through 4.15 and [16, Theorem 5.15] we obtain the following characterization when F is of type II * : Theorem 4.16.If Y contains a fiber F of type II * and B .= π(F ) is not a triple conic, then P is unstable.4.4.The stability of P when F is of type III * .We now consider the case when F is of type III * .From[16, Theorem 5.16] the curve B can only be realized by one of the following is semistable there are two possibilities: either C is smooth, in which case P is stable (Corollary 4.8.1); or C is singular and then P is strictly semistable if and only if there exists a one-parameter subgroup λ (and coordinates in P 2 ) such that 2C and B are both non-stable with respect to this λ (Theorem 4.11).When B is unstable we can use the explicit constructions obtained in[16]to conclude P is strictly semistable.Proposition 4.17.If Y contains a fiber F of type III * and B .= π(F ) consists of a triple line, a double line and another line in general position, then P is not stable.Proof.Let P and Y be as above.One can find coordinates in P 2 as in[16, Example 7.49] so that the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (3) of Theorem 2.7 and we conclude P is not stable.Alternatively, we can also apply Theorem 3.3.Lemma 4.18.If P contains a curve B and a base point P such that mult P (B) = 6, then P is not stable.Proof.Since mult P (2C) ≥ 2, the result follows from [17, Theorem 1.3].Proposition 4.19.If Y contains a fiber F of type III * and B .= π(F ) consists of a triple line, a double line and another line concurrent at a base point, then P is not stable.Proof.Let P, Y and B be as above.Then P contains a base point P (the point where the 3 lines meet) such that mult P (B) = 6, and the result follows from Lemma 4.18.Proposition 4.20.If Y contains a fiber F of type III * and B .= π(F ) consists of a double line, a nodal cubic and another line, then P is not stable.Proof.Let P and Y be as above.One can find coordinates in P 2[16, Example 7.45] so that the Plücker coordinates of P with respect to these coordinates satisfy the conditions in Case (3) of Theorem 2.7 and we conclude P is not stable.Proposition 4.21.If Y contains a fiber F of type III * and B .= π(F ) contains a line with multiplicity four, then P is not stable.Proof.If B contains a line with multiplicity four, then we can find coordinates in P 2 and generators of P which are given by equations as in Case 1 of Theorem 3.1.Proposition 4.22.If Y contains a fiber F of type III * and B .
. It follows from Theorem 3.4.Theorem 4.35.If Y contains a fiber of type IV * and P is not stable, then C is singular and B is semistable.Proof.If P is not stable, then it follows from Corollary 4.8.1 that either C is singular or B is unstable.Now, if B is unstable, then the results from [16, Sections 6 and 7] and [15, Section 2] together with Theorem 4.33 tell us B can only (possibly) be (1) a double line, a nodal cubic and another line ([16, Example 7.35]) or (2) two double lines and a conic ([16, Example 7.38]) but in any case P is stable by Lemma 4.34.