Below are the notes I wrote in the 90's about self maps of P1 ******************************************************************** 0) Below is a treatment of compactification and stability for self maps of P^1. 1) We are interested in maps f: P^1 -> P^1 of degree d > 0 modulo the conjugation action. We first restrict ourselves to the d even case. Later, in (7), I will include some comments on the odd case. To start, I will reinterpret Silverman's calculation in a more convenient geometric framework --- it is not necessary to read his paper to follow the discussion below. Let V be a 2-d complex vector space and let P^1=P(V). Let Z be the projective space of sections of the line bundle O(d,1) on P(V) x P(V), Z = P( H^0(P(V) x P(V),O(d,1)) ). Let SL(V) act on P(V) x P(V) by g * (p,q) = (g * p , g * q). There is a canonically induced action of SL(V) on Z. If Z is viewed as graphs of degree d maps from the first P(V) to the second P(V), then action of SL(V) on a function f:P(V)-> P(V) is g * f = g o f o g^{-1} which is the standard conjugation action. Z is a 2d+1 - dimensional projective space, and the above action of SL(V) on Z is exactly the action analyzed in Silverman's paper . Both are, as projective representations of SL(V), P(Sym^d(V*) \tensor V*). However, we will see viewing the action geometrically will lead in a different direction. In the above situation, stability has a simple geometric interpretation. Let [f]\in X where f is a curve in the linear series O(d,1) on P(V) x P(V). The curve f must consist of a union of one horizontal section h with a collection of possibly multiple vertical sections v_1,..., v_k. Let D be the diagonal in P(V) x P(V). [f] \in X is GIT stable if and only if the following two conditions hold: (i) there does not exists a vertical section of multiplicity > (d+1)/2. (ii) If v is a vertical section of multiplicity > (d-1)/2 then v intersect h does not lie in D. Also not stable <=> unstable. Note the above condition (ii) is SL(V) invariant since D is invariant. To prove the stability statement, we use the numerical criterion. These calculations are simple (and, anyway, appear in Silverman's paper). Similar statments for d=odd exist (with a non-empty semistable locus). 2) Up to now, I have just reinterpreted the GIT calculation. Here is the new twist which I think makes the geometric framework much more interesting (to me at least). We will consider a new projective variety Y with an SL(V)-equivariant map j:Y -> Z. Y will be the moduli space of stable maps from genus 0, n-pointed curves to P(V) x P(V) representing the class (d,1) in P(V) x P(V). Formally, Y= \bar{M}_{0,n}( P(V) x P(V) , (d,1) ) Perhaps, some further explanation: (a) Let C be a connected reduced nodal curve of arithmetic genus 0 a tree of P^1's) with n marked nonsingular points. (b) A map f: C -> P(V) x P(V) is stable if and only if every component of C that is collapsed by f to a point contains at least 3 special points of C (nodes or markings). (c) The map f represents the curve class (d,1) if f_*[C]= (d,1). The morphism j:Y -> Z is obtained by associating to [f] in Y the cycle (with multiplicities) [ f_*[C] ] in Z. The SL(V) action on P(V) x P(V) considered above induces an SL(V) action on Y. That is, g * [f] = g o f: C -> P(V) x P(V). Now, here is the first point. Since Z^{s}= Z^{ss} in part (1), it is not hard to find a linearization on Y such that Y^{s}= Y^{ss} = inverse image of Z^{s} = j^{-1}(Z^{s}) . (To find such a linearization, proceed as follows: Let L be O(1) on Z. Then, j^*(L) is a linearized line bundle on Y (though not ample on Y). Let N be any SL(V)-linearized ample line bundle on Y. Then the linearization we want is j^*(L)^{999999999999999999999999} \tensor N . I can explain this better.) We now have a compact GIT quotient M = Y^{s} // SL (V). 3) We now examine this quotient point theoretically. Since Y has orbifold singularities and SL(V) acts on Y with finite stabilizers, we can, with some work, conclude M has only orbifold singularities. M is nonsingular Deligne-Mumford stack. The points of M parameterize stable self maps: Let d be even and positive. Let C be a tree of P^1's with n nonsingular markings. (a) A self map of degree d is a map f: C -> C where Im(f) lies in one component P^1 (unique since d>0). The degree is defined by the obvious summation of the degrees of the domain components. (b) Two self maps f:C-> C, f':C'->C' are equivalent if there is an isomorphism g: C -> C' which respects the markings and satisfies: g o f o g^{-1} = f' . (c) Let I be the unique image component of a self map f: C -> C. Let I_{ord} be the set of ordinary points (non-special) of I. Let W_1,...., W_r be the connected components of C \ I_{ord}. Let e_i be the degree of f on W_i. Let p_i be the unique point at with W_i attaches to I. (d) A self map f: C -> C is stable if and only if the following hold: (i) If J not equal I is a component of C collapsed to a point by f, then J contains at least 3 special points of C (nodes or markings). (ii) For all i, e_i < (d+1)/2 (iii) If e_i > (d-1)/2, then f(p_i) is not equal to p_i. To show M = Y^{s} // SL(V) exactly parametrizes these stable self maps, we just examine the points of Y^{s}=j^{-1}(Z^{s}). Let [f]\in Y^{s} where f: C -> P(V) x P(V). Since f_*[C] is of class (d,1), there is a unique component I of C such that f_*[I] is of class (k,1) for some k. That is, the image of I is the unique horizontal section h of f_*[I]. Moreover f: I-> h is a birational map of nonsingular P^1's, hence an isomorphism. Let q1, q2 be the projections of P(V) x P(V) on the first and second factors respectively. Then, t: I -> P(V) defined by t = q1 o f is an isomorphism. The main claim is that F: C -> I, F = t^{-1} o q2 o f, is a stable self map canonically associated to the point [f]\in Y^{s}. Condition (i) of self map stability is inherited from the map stability of f: C -> P(V) x P(V). Condition (ii) is obtained from the first part of the GIT stability statement. Condition (iii) is obtained from the second part of the GIT stability statement. Suppose g is in SL(V). I claim the stable self map associated to g * [f] \in Y^{s} is equivalent to the self map associated to [f]. Let [f']= g * [f]. Then, f': C -> P(V) x P(V) is simply f g the composition C -> P(V) x P(V) -> P(V) x P(V) . The distinguished component I is the same as before. The new isomorphism t':I -> P(V), t' = q1 o f', is related to t by g o t = t' The new map F' = t'^{-1} o q2 o f' = = t^{-1} o g^{-1} o g o q2 o f = F. So the orbits of SL(V) all correspond to the same stable self map. The reverse construction is the following. Suppose F: C -> I \subset C is a stable self map of degree d. There is a canonical projection p: C -> I by mapping I -> I by the identity and mapping each connected component W_i of C \ I to the connection point p_i. Let f: C -> I x I be given by f= (p,F). Select any isomorphism I -> P(V) (SL(V) choice!). Then f: C-> P(V) x P(V) is obtained where [f]\in Y^{s}. These constructions are inverses to each other (up to SL(V) orbit). 4) For d even, we now have a geometric compactification M which parametrizes stable self maps with n marked points. (i) M is an irreducible compactification of the open problem, M has orbifold singularities, and the boundary at infinity is a divisor with normal crossings. (ii) M is equipped a universal family, F -> M, a universal self map F -> F_B -> M (there is a twist here as the universal self map is to a blow-down F_B of the universal family), and canonical cotangent line bundles. (iii) M is equipped with another set of very interesting line bundles associated to each marking. Let i be a fixed marking (one of the n markings). On the prequotient, Y^s, we can consider evaluation maps ev_i: Y^s -> P(V) x P(V). Then, ev_i^* ( D ) is an SL(V) equivariant divisor, and hence defines a divisor class D_i on the quotient M. Recall, D in P(V) x P(V) is the diagonal. The meaning of D_i : it is the locus where the i^th marking is a fixed point of the self map. One may now set up an integral theory for the moduli of self maps. The integrands will be all possible combinations of the cotangent line classes and the classes D_i. Roughly cotangent lines correspond to imposed vanishing of the differential --- critical point. If both cotangent line and D_i conditions are imposed, one should get loci where the marking is a critical fixed point (up to boundary corrections). All of these questions are natural from the point of view of the compactification, and in some sense should have been pursued years ago. 5) I will make some remarks on the d odd case in (7) below. In order to avoid semistable issues, which I really do wish to avoid, I will use some sleight of hand. In fact, the odd case will covered by a more general framework which is natural (and in fact inevitable) from the point of view of the d even case studied above. Let d positive. Let d_1, ..., d_n be fixed non-negative weights such that: d_T = d + \sum_{i=1}^{n} d_i = 0 mod 2 . I will now define a moduli space, M(d | d_1,...,d_n), of stable degree d self maps with n weighted markings (weighted by d_1, ..., d_n), Let C be a tree of P^1's with n nonsingular markings. (a) A self map of degree d is a map f: C -> C where Im(f) lies in one component P^1 (unique since d>0). The degree is defined by the obvious summation of the degrees of the domain components. (b) Two self maps f:C-> C, f':C'->C' are equivalent if there is an isomorphism g: C -> C' which respects the markings and satisfies: g o f o g^{-1} = f' . (c) Let I be the unique image component of a self map f: C -> C. Let I_{ord} be the set of ordinary points (non-special) of I. Let W_1,...., W_r be the connected components of C \ I_{ord}. Let e_i be the degree of f on W_i. Let deg_i be the sum of the weights of the marked points on W_i Let p_i be the unique point at with W_i attaches to I. (d) A self map f: C -> C is stable if and only if the following hold: (i) If J not equal I is a component of C collapsed to a point by f, then J contains at least 3 special points of C (nodes or markings). (ii) For all i, e_i + deg_i <= (d_T+1)/2 . (iii) If e_i + deg_i > (d_T-1)/2, then f(p_i) is not equal to p_i. The first moduli space of degree d even self maps with n marked points discussed in (3)-(4) is a special case of the above construction: that space is simply M(d | 0,...,0), where all the weights are taken to be 0. I claim the moduli of stable degree d self maps with n weighted markings, M(d | d_1,...,d_n), is again perfectly well behaved with all the properties of the even case holding: irreduciblity, orbifold singularities, normal crossing boundary, universal objects, and tautological classes. Note now the tautological classes are atteched to weighted points. Again, the integral theory is ready to be studied. We note if the weights are too high (for example if d_i > d_T/2 for some i), the the moduli space is empty. 6) The construction of M(d | d_1,...,d_n) is straightforward using the ideas above. Let Y as before the moduli Y= \bar{M}_{0,n}( P(V) x P(V) , (d,1) ) Let Z now be the space of sections of O(d_T,1) on P(V) x P(V): Z= P( H^0( P(V) x P(V) , (d_T,1)) ). A new morphism j:Y -> Z is obtained by associating to [f] in Y the cycle (with multiplicities) [ f_*[C] ] + \sum_{i=1}^n d_i [v(q1(f(i)))] in Z. Here q1(f(i)) is the first coordinate of the f evaluation of the i^th marked point. And, v(p) for p in P(V) is the vertical curve given by p x P(V). The SL(V) action on P(V) x P(V) considered above induces an SL(V) action on Y. That is, g * [f] = g o f: C -> P(V) x P(V). Now, we repeat as before. Since Z^{s}= Z^{ss} in part (1), there exists a linearization on Y such that Y^{s}= Y^{ss} = inverse image of Z^{s} = j^{-1}(Z^{s}) . Then, let M(d | d_1,...,d_n) = Y^{s} // SL (V). A point theoretic analysis as before then identifies the points of the quotient with stable degree d self maps with n weighted markings. 7) Let d be positive and odd. We may now consider odd self maps. The simplest is the case is M(d | 1,0,..,0), with 1 point of weight 1 and the rest of weight 0. The moduli space M(d | 1,0,...,0) has an open set which is exactly degree d self maps with 1+n distinct points. The case d=1 is special since then the marking of weight 1 forces condition (iii) of stability to hold: the open set is then degree 1 self maps which move the point of weight 1. It more natural to consider all the M(d | d_1,...,d_n) together. The reason is that the the various moduli space M(d | d_1,...,d_n) occur as boundary strata in each other. In fact, the recusive geometry of the boundary strata will be the key for the integral study: perhaps leading to a full determination. I know some more about this, but enough for now.