On the cohomology of certain subspaces of $\mathit{Sym}^n(\P^1)$ and Occam's razor for Hodge structures

In \cite{Vakil13} Vakil and Wood made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of $\Sym^n(\P^1)$. In this note, we disprove one of them thereby obtaining a counterexample to the principle of Occcam's razor for Hodge structures; and we prove that the other conjecture, with a minor correction, holds true.


Introduction
For a smooth and proper variety X over , the Hodge-Deligne polynomial determines the Hodge numbers; but that is no longer the case when X is not smooth and proper. To elaborate, for any variety X over , the compactly supported cohomology groups H i c (X , ) carry Deligne's mixed hodge structures. One defines the Hodge-Deligne polynomial as When X is smooth and proper, one has e p,q = (−1) i h p,q (H i (X , )). There are many examples where the simplest possibility holds i.e. there is a simplest Hodge structure on H i c (X , ) for all i in agreement with the virtual Hodge structure. In [VMW13], Vakil and Wood dub this wellknown principle as "Occam's razor for Hodge structures". This principle led them to conjecture about the stable rational cohomology of certain subspaces of Sym m ( 1 ), the m-fold symmetric product of 1 . These are Conjectures G' and H' in [VMW13] 1 . The goal of this note is to disprove one of them, and prove a slight alteration of the other.
Before stating their conjectures and the main theorem of this note, we fix some notations. All varieties are over . For a complex quasiprojective variety X , let Sym m (X ) denote the n-fold symmetric product. For a partition λ of m and a complex variety X , let w λ (X ) denote the locally closed subset of Sym m (X ) with multiplicities precisely λ. Further more, let UConf n X denote the unordered configuration space of n points on X i.e. UConf n X = w 1 n (X ), and let PConf n X denote the ordered configuration space of n points on X .
Conjecture H' of [VMW13] states that the values of i > 0 for which Our goal is to prove the following theorem.
Theorem A. Let n ≥ 2. Then otherwise.
In particular, for all i ≥ 1, The following corollary to Theorem A disproves Conjecture G'.
Remark 1. A question along the lines of the conjectures based on the Occam's razor of Hodge structures would be, can one determine the rational cohomology of a variety over by counting the number of q points of that variety? The answer, in general, is in negative. In fact, the conjectures were made on the basis of such point-counts. The Grothendieck-Lefschtez trace formula (see [Gro66]) allows one to count the number of q points of a variety X from its topology, whenX is a reasonably nice variety. However, there is no sufficient criterion to cross the bridge from q points of a variety to the rational Betti numbers of the topological space formed by its -points. This note provides two such examples.
I am grateful to my advisor, Benson Farb for his helpful comments and patient guidance. I also deeply thank Melanie Matchett-Wood for her valuable feedback on an earlier draft of the manuscript.
2 Cohomological stability of some locally closed strata of Sym m ( 1 ) In this section we prove Theorem A and Corollary B. One of the important steps is to compute H * (UConf n × ; ). The latter quantity is well-known (see e.g. [Sch18], [DCK17] and the references therein); in this paper, we will heavily use the notion of spaces admitting a semi-filtration developed in [Ban19] to give a short alternative method to compute H * (UConf n × ; ). Our proof of Theorem A can be outlined via the following steps.
1. Describe the space w 1 n 22 ( 1 ) is a fibre-bundles over UConf 2 ( 1 ) with fibres isomorphic to UConf n ( × ).  . Put X = × in (2.1). Then for p ≥ 1, the spectral sequence (2.1) gives us: and for p = 0 one has with the differentials going horizontally E p,q 1 → E p+1,q 1 (see Figure 1). Furthermore, one can read off the weights from the explicit description of the terms E p,q 1 of the spectral sequence in (2.2) by noting that H 1 c ( × ; ) is pure of weight −2 and Hodge type (−1, −1). Letting (1) denote the Tate Hodge structure of weight −2 and Hodge type (−1, −1), we obtain: and note that for all {a, b} ∈ UConf 2 ( 1 ), An equivalent description of w 1 n 22 ( 1 ) is that it's a quotient of the fibre bundle by the action of S n on the fibres of F 2 , where and the action of S 2 on the base PConf 2 ( 1 ). Let P B n := π 1 (PConf n ( 1 )) be the pure Hurwitz braid group on n strands. Then P B 2 acts trivially on the homology of the fibres of F 2 because it acts by conjugation on π 1 (UConf n ( × )). Therefore, the Hurwitz braid group on two strands π 1 (UConf 2 ( 1 )) also acts by conjugation on π 1 (UConf n ( × )), so the monodromy is trivial on the homology of the fibres of π 22 in (2.4). On the other hand H * c (UConf 2 ( 1 ); ) ∼ = . One way to see this is by using the longexact sequence of cohomology. Equivalently, plugging X = 1 and n = 2 in (2.1) (see [Ban19, Corollary 2]), we get the following spectral sequence: To this end, we think of Sym 2 ( 1 ) as the space of degree 2 divisors in 1 ; so we have an isomorphism given by a global section in 1 (2) mapping to its divisor (for a vector space V , we denote its dual by V ∨ ). Note that (Γ ( 1 , 1 (2)) ∨ ) ∼ = 2 . In terms of coordinates one can write down F −1 as: Now F −1 • ∆ : 1 → 2 embeds 1 as a smooth conic in 2 ; it is the discriminant locus cut out by y 2 − xz = 0. And observe that is an isomorphism for i = 0, 2. Indeed, the fundamental class of 2 restricts to that of 1 for i = 0, and for i = 2 the hyperplane class in H 2 ( 2 ; ) restricts to twice the class of a point in 1 by Bézout's theorem, thereby inducing an isomorphism on cohomology with coefficients in degree 2. Combining this information with the spectral sequence in (2.5) we get that The Serre spectral sequence for the fibration (2.4), combined with (2.2), (2.8), and the fact that π 1 (UConf 2 ( 1 )) acts trivially on H * (UConf n ( × ); ), gives us Therefore for all i ≥ 0 we have proving Theorem A.
Proof of Corollary B. The space w 1 n 23 ( 1 ) is a fibre bundle: . An equivalent description of w 1 n 23 ( 1 ) is that it's a quotient of the fibre bundle