Stable cohomology of the universal degree $d$ hypersurface in $\mathbb{P}^n$

Let $U_{d,n}^*$ be the universal degree $d$ hypersurface in $\mathbb{P}^n$. In this paper we compute the stable (with respect to $d$) cohomology of $U_{d,n}^*$ and give a geometric description of the stable classes. This builds on work of Tommasi and Das .


Introduction
Let U d,n be the parameter space of smooth degree d hypersurfaces in P n .There is a natural inclusion U d,n ⊆ P ( n+d d ) = P(V d,n ), where V d,n is the vector space of homogenous degree d complex polynomials in n + 1 variables.Let be defined by φ(f, p) = f .The map φ : U * d,n → U d,n is the universal family of smooth degree d hypersurfaces in P n ; it satisfies the following property: given a family π : E → B of smooth degree d hypersurfaces in P n there is a unique diagram: In other words, any family of smooth degree d hypersurfaces is pulled back from this one.Our main result is as follows: Theorem 1.1.Let d, n ≥ 1.Then there is an embedding of graded algebras: (1) The element φ(x) = c 1 (L ) where L is the fiberwise canonical bundle(defined in Section 2).(2) Suppose d ≥ 4n + 1.Then φ is surjective in degree less than d−1 2 .We now define some spaces related to U d,n , where we will prove similar results.Let X d,n ⊆ V d,n be the open subspace of polynomials defining a nonsingular hypersurface.The complement of X d,n in V d,n is known as the discriminant hypersurface; it is the zero locus of the classical discriminant polynomial.It is known to be highly singular.
A point of X d,n determines a projective hypersurface up to a scalar.There is a natural action of C * on X d,n .Let U d,n = X d,n /C * be the quotient of this action.
We can further quotient to obtain M d,n := U d,n /P GL n+1 C, the moduli space of degree d smooth hypersurfaces in P n .Let X * d,n := {(f, p)|f ∈ X d,n , p ∈ P n , f (p) = 0}.There is a forgetful map π : X * d,n → X d,n defined by π(f, p) = f .The fibres of π are Z(f ) := π −1 (f ) = {p ∈ P n |f (p) = 0} ⊆ P n .It is well known that the map π is a fibre bundle.
X * d,n also has several interesting quotients.The action of GL n+1 on X d,n lifts to one on X * d,n .We obtain U * d,n = X * d,n /C * .The map π : X * d,n → X d,n is C *equivariant and descends to the map φ : U * d,n → U d,n .We also have We can rewrite our result in terms of X * d,n and M * d,n as well.This is important to us as our proof will mostly involve understanding the space X * d,n .The space (1) There is an embedding of graded algebras: (2) There is an embedding of graded algebras: Suppose that d ≥ 4n + 1.Then, the maps ψ and ϕ are surjective in degree ≤ d−1 2 .Theorem 1.2 is equivalent to Theorem 1.1 after applying Theorem 2 of [5].Nature of stable cohomology: Throughout the course of the proof of Theorem 1.2 we also obtain the following description of the stable cohomology classes of X * d,nthe stable classes are tautological in the following sense: There is a line bundle L on M * d,n defined by taking the canonical bundle fibrewise (we rigorously define L in Section 2).We will show that c 1 (L ), . . ., c 1 (L ) n−1 are nonzero in H * (M * d,n ; Q) and that stably the entire cohomology ring of M * d,n is just the algebra generated by c 1 (L ).By [5], In this way we have some qualitative understanding of the stable cohomology of Both the statement of Theorem 1.2 and our proof of it are heavily influenced by [6], in which Tommasi proves analogous theorems for X d,n .Our techniques and approach are also similar to that of Das in [1],where he proves In some sense, this paper shows that in a stable range, something similar to Das's theorem is true for marked hypersurfaces in general.Method of Proof.One could attempt to prove Theorem 1.2 by applying the Serre spectral sequence to the fibration π : X * d,n → X d,n .To successfully do this however, one would need to understand the groups H p (X d,n ; H q (Z(f ); Q)).While we do a priori understand what the groups H p (X d,n ; Q) are (This is the main theorem of [6]), this is not sufficient for us to understand what the groups H p (X d,n ; H q (Z(f ); Q)) are, since H q (Z(f ); Q) is a nontrivial local coefficient system.Instead we use an idea of Das and compute H * (X p d,n ; Q), where X p d,n := {f ∈ X d,n |f (p) = 0} to avoid any computations with nontrivial coefficient systems.After we have proved Theorem 1.2 we can use it to deduce what these twisted cohomology groups are.
in the stable range.More precisely, we have the following: Proposition 2.1.Let n ≥ 0, and let d > n + 1.Then there is a natural embedding.
The inclusion is one of algebras.
Proof.We first define the fiberwise canonical bundle L over X * d,n as follows: Note that the bundle L is actually pulled back from a bundle on M * d,n := X * d,n /GL n+1 (C).By the same argument as in Theorem 1 of [5], n be an orbit map.More precisely, Theorem 1 of [5] states that the natural map with a basis given by some set {α i } such that the pullbacks {i * (α i )} give a basis of where ω Z is the Kahler class of the variety Z.This implies that for 3. The space X p d and the Vassiliev method Let us define the ordered and unordered configuration space of a space X as we will need to consider thes in this section.Given a space X, the nth ordered configuration space of X denoted PConf n X is There is a natural action of the symmetric group on n letters S n on X by permuting the coordinates.The quotient PConf n X/S n is called the nth unordered configuration space and denoted UConf n X.In order to understand X d,n we will first look at the cohomology of a related space.For a fixed point p ∈ P n we set d is a vector space.The complement of X p d in V p d will be called Σ d,p .We will compute its Borel-Moore homology and use Alexander duality to compute In what follows we will often refer to a group G p defined as follows: if p ∈ P n , it is by definition a one-dimensional subspace p ⊆ C n+1 .Choose a complementary subspace W ⊆ C n+1 (it is not unique, but we will fix a particular one).We let The group G p acts on P n fixing p. Therefore it acts on both X p d and C n −{0}.The the map π is equivariant with respect to these actions.The map π therefore is the pullback of a map from π ′ : We will try to understand the Borel-Moore homology of Σ v .
To accomplish this the Vassiliev method [8] will be applied.The Vassiliev method to compute Borel-Moore homology involves stratifying a space and using the associated spectral sequence to compute its Borel Moore homology.The space Σ v will be stratified based on the points at which a section f is singular.The techniques used are very similar to that in [6] which contains many of the technical details.
We will now construct a cubical space X which will be involved in understanding Σ v .Let N = d−1 2 .Let I be a subset of {1, . . ., N − 1}.For k < N , let We define If I ⊆ J then we have a natural map from X J → X I defined by restricting p.This gives X • the structure of a cubical space over the set {1, . . ., N }.We can take the geometric realization of X • denoted by |X|.Then there is a map ρ : |X| → Σ v , induced by the forgetful maps |X| is topologised in a non-standard way so as to make ρ proper.We topologise it as follows: in [6] Proof.This proof is nearly identical to that of Lemma 15 in [6].The properness of ρ : Y → Σ v follows from the properness of ρ : |X | → Σ and the properties of the subspace topology.In our setting having contractible fibres implies that the map ρ is a homotopy equivalence, this follows by combining Theorem 1.1 and Theorem 1.2 of [4].We will now prove that the fibres are contractible.If f ∈ Σ≥N v , let {p 1 , . . ., p k } be the singular zeroes of f .In this case the fibre ρ −1 (f ) is a simplex with vertices given by the images of the points (f, Now as in any geometric realization, |X| is filtered by The F n form an increasing filtration of |X| , i.e.
We have a map φ : The map φ expresses F k − F k−1 as a fibre bundle over B k with ∆ • k fibres, i.e we have a diagram as follows: We have a map B k → UConf k (P n − p) defined by {f, p 0 . . ., p k } → {p 0 , . . .p k }.This is a vector bundle by Lemma 3.2 in [7].
We have a one-dimensional local coefficient system denoted ±Q on UConf k (P n − p) defined in the following way: Let S k be the symmetric group on k letters.We have a homomorphism π 1 UConf k (P n − p) → S k associated to the covering space PConf k (P n − p) → UConf k (P n − p).Compose this homomorphism with the sign representation S k → ±1 = GL 1 (Q) to obtain our local system.As with any filtered space, we have a spectral sequence with E p,q 1 = Hp+q (F p − F p−1 ; Q) converging to H * (Y ; Q).Now for p < N by Proposition 3.4 E p,q 1 = Hq−(2e d −2(n+1)(p+1)) (UConf p (P n − p); ±Q).We would like to claim that E N,q 1 doesn't matter in the stable range.To be more precise, we have the following: Proof.We first will try to bound the H * (F N ; Q) and then use the long exact sequence of the pair.F N is the union of locally closed subspaces We have a surjection π : φ k → UConf k (P n − p).This map π is in fact a fibre bundle with fibres ∆ k × C e d −N (n+1) .The space UConf k (P n − p) is kn dimensional.Therefore

This implies that for all k, H * (φ
By the long exact sequence in Borel Moore homology associated to the pair

Interlude
In [6], Tommasi proves the following result: In this section we shall look at the proof of Theorem 4.1 in [6] and use it to prove an identity used later on in this paper.One of the ingredients in the proof of Theorem 4.1 is a Vassiliev spectral sequence.We introduce a new convention, by letting h denote the dimension of H.We also define Gr(p, n) to be the Grassmanian of p-planes in C n .In what follows we shall need a few basic facts about H * (Gr(p, n); Q) and Schubert symbols.Let be a complete flag.Given U ∈ Gr(p, n), we can associate to it a sequence of numbers, a i = dimU ∩ E i .These a i satisfy the following conditions: Such sequences are called Schubert symbols.Let a = (a 0 . . .a n ).We call a a Schubert symbol if 0 ≤ a i+1 − a i ≤ 1,a 0 = 0 and a n = p.Associated to each Schubert symbol a we have a subvariety W a ⊆ Gr(p, C n ) defined as follows.We define The main result we will be using is the following.
The last equality follows because if we are given a sequence of a i , we can uniquely obtain a sequence of b i , by letting b i = a i − a i−1 .
Our main aim of this section is to prove the following technical result that is necessary for our purposes.
Theorem 4.4.The Vassiliev spectral sequence in [6] degenerates in the stable range: if p < d+1 2 and if q > 0, then E p,q (These are the diagonal terms in the spectral sequence).
Proof.We already know that because the left hand side of (1) are the appropriate terms in a spectral sequence converging to the right hand side of (1).
It suffices to prove that Lemma 2 in [8] states that: By Proposition 4.3 this is equal to 2 n+1 .

Computation
We would like to know what the groups H * (UConf k+1 (P n − p); ±Q) are.First note that in [8] Vassiliev proves that : Also note that in light of Theorem 4.2 the homology of Grassmannians is well understood in terms of Schubert cells.
Consider the long exact sequence in Borel Moore homology associated to The last inclusion is defined by the map φ : UConf We obtain a long exact sequence as follows: (2) 2 , then H k (X v ) ∼ = H k (G p ) as vector spaces.Proof.Now in our spectral sequence we had E p,q 1 = Hq−(2e d −2(p+1)(n+1)) (UConf p+1 (P n − p); ±Q).First collect all terms in the main diagonal, i.e.
We have another equality from Proposition 5.2, Plugging this into (4) we have We have the identity This implies, Now we will try to prove 3 by induction on k.For k = 0, (3) is trivial.By induction

Putting this into 5 we obtain
Now we can look at the Serre Spectral sequence associated to the fibration We observe that if there are no nonzero differentials, then 2n−1 .This is because the Serre spectral sequence degenerates and since Q[e 2n−1 ]/e 2 2n−1 is a free graded commutative algebra the ring structure of the total space is forced to be the tensor product.Proposition 5.5.Let d > 0 and p ∈ P n .Then, , where A is H * (X p d /G p ; Q).Proof.This follows immediately from Theorem 2 in [5].
We will also need the following fact that is a special case of Lemma 2.6 in [1].
, where |e 1 | = 1.Proposition 5.6 implies if there are no nonzero differentials in both our Vassiliev spectral sequence and in the Serre spectral sequence associated to the fibration In case there are nonzero differentials in either spectral sequence, then

Comparing fibre bundles
In this section we finish the proof of Theorem 1.2.
Proof of Theorem 1.2.We compare three related fibre bundles and their associated spectral sequences.This is similar to the Proof of Theorem 1.1 in [1].
(7) P G p := Stab P GL(n+1) p By Proposition 5.4 and Theorem 1 of [5] there are two possibilities for H * (U d,p ; Q): either

Suppose for the sake of contradiction that H
Then since the homology of the base and the fibres are isomorphic, 2 .However by Proposition 2.1, So we must be in the case where, /(e 2 2n−1 ).Consider the trivial fibre bundle U d × P n → P n .There is a natural inclusion of fibre bundles as shown in (7).This induces a map of spectral sequences between the associated Serre spectral sequences.
Note that any class α ∈ H q (U p d ; Q) that lies in the image of H q (U d ; Q) is mapped to zero under any differential thanks to the fact that all dfferentials are zero in the spectral sequence associated to a trivial fibration.The only possible nonzero differential in the E 2 page of the Serre spectral sequence associated to the fibration U * d → P n is d(e 2n−1 ).Suppose for contradiction that d(e 2n−1 ) = 0.This implies that .If d ≥ 4n + 1, then this implies that (1 + t 2n+1 ) |p(t).This is a contradiction.So we must have a differential killing the class in H 2n (P n , H 0 (U d,p )); Q).The differential must come from from e 2n−1 , i.e. d(e 2n−1 ) = ax n for some a ∈ Q * .This (along with multiplicativity of differentials) determines all differentials and implies (1).By Proposition 5.6 (1) =⇒ (2).By Theorem 1 of By Theorem 1.2, we know that the classes in the E 2 page corresponding to the group H p (GL n+1 (C); c 1 (L ) q ) survive till the E ∞ page and in the stable range all other terms are killed by differentials.Now suppose n is even.Then the only other terms in the spectral sequence are of the form H p (X d ; H n−1 (Z(f ); Q)).However it is not possible for any such term to be in the image or in the preimage of a nonzero differential.This is because all other terms survive so any possible nonzero differential must be from H p1 (X d ; H n−1 (Z(f ); Q)) to H p2 (X d ; H n−1 (Z(f ); Q)) for some choice of p 1 and p 2 .However no differential is of bidegree (p 2 − p 1 , 0).This implies that H p (X d ; H n−1 (Z(f ); Q)) ∼ = 0.
A similar argument shows that if n is odd, H p (X d ; H n−1 (Z(f ); Q)) ∼ = H p (X d ; Q).Essentially the only difference between the even case and the odd case is that in the odd case we have a class c 1 (L ) local coordinates in a neighbourhood U containing p. Pick a local trivialisation s of the line bundle O(d) in U .There is an induced map f * : T * 0 (O(d) p ) → T * p (P n ).Let us use our local coordinates to identify T * 0 (O(d) p ) with C and T * p (P n ) with C n .Suppose f ∈ X p d .Then the map f * is nonzero because f has a regular zero locus.Let This defines a map , a space |X | is constructed with a map ρ : |X | → Σ.Here, Σ = V d − X d .The topology on |X | is chosen carefully so as to make ρ proper.The construction of |X | as a set identical to that of |X| except we replace Σ v with Σ.There is a natural inclusion |X| → |X |.We give |X| the subspace topology along this map.Proposition 3.2.The map ρ : |X| → Σ v is a proper homotopy equivalence.
Here Q(σ) is the local sytem obtained by the action of π 1 (UConf k (P n − p)) on the fibres Hk (∆ • k ) where in this case ∆ circ k is the open k simplex corresponding to the fibres of the map F k − F k−1 → B k .But one observes that the action of π 1 (UConf k (P n − p)) on this open simplex is by permutation of the vertices which imples that Q(σ) = ±Q.

Theorem 4 . 2 .
Let a be a Schubert symbol.The classes [W a ] ∈ H * (Gr(p, n); Q) form a basis.For a proof of Theorem 4.2 see page 1071 of [3].Proposition 4.3.Let n be a positive integer.Then k,p