Symmetric Powers Do Not Stabilize

We discuss the stabilization of symmetric products Sym^n(X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h^0(X, \omega_X), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain"motivic zeta function"associated to a curve, and its Hodge polygon.


Introduction
Let k be a field, and K 0 (Var k ) the Grothendieck ring of varieties over k. This is the free abelian group on isomorphism classes [X] of separated, finite type k-schemes (varieties), subject to the following relation:  [20,Conjecture 1.25] that in a certain completion of K 0 (V ar k )[L −1 ], denotedK (to be defined in Section 2), the limit lim n→∞ [Sym n X] L n dim(X) exists for connected X. They call the existence of this limit "motivic stabilization of symmetric powers" or MSSP for short; they show this conjecture is true in many cases. The main goal of this note is to provide some evidence that it is false in general.
We introduce analogous but more accessible claims (False Claims 4 and 5) which hold true in every case where MSSP is known to hold. Furthermore, we show unconditionally that both False Claims are untrue in general for k = C-counterexamples include smooth projective surfaces X with geometric genus p g (X) = 0 (Corollary 21). 1 We show that these counterexamples are also counterexamples to MSSP, conditional on the truth of either of two well-known conjectures about K 0 (Var k ) (the "Cut-and-Paste" conjecture of Liu and Sebag [13] or the conjecture that L is not a zero divisor), and thus MSSP is false if either of these conjectures are true (Corollary 23).
Finally, we propose Hodge-theoretic heuristics explaining the failure of MSSP, and give some evidence for these heuristics via explicit computations for curves. In particular, we prove a motivic analogue of the classical theorem that "the Newton polygon lies above the Hodge polygon," for Weil zeta functions associated to varieties. Namely, if X is a smooth projective curve with a rational point, we show that over a general field, a certain "motivic Newton polygon" associated to X lies above the Hodge polygon of X; this implies the classical result for Weil zeta functions if X is defined over a finite field (Lemma 29). Over algebraically closed fields of characteristic zero, we show that the Newton and Hodge polygons of curves are 1 While this paper was in preparation, Melanie Wood provided several other counterexamples to False Claim 4 [22]. In particular, her arguments combined with those here show that for smooth projective surfaces with a non-vanishing even plurigenus, False Claim 4 fails; in addition to the results here, this covers e.g. Enriques surfaces and certain surfaces of general type but with pg = 0. equal (Corollary 31). These last results are of independent interest, and are contained in Section 5, which can be read independently of Sections 3 and 4.

False Claims and Motivation
Let us first give the statement of MSSP. We define a filtration on K 0 (Var k )[L −1 ], given by dimension. Let We define the ringK to be the completion of K 0 (Var k ) at this filtration, that is, The ringK was initially defined by Kontsevich [9] as the ring in which the values of motivic integrals lie. The MSSP Conjecture takes place inK: exists inK.
Remark 2. Note that resolution of singularities implies that F i K 0 (Var k ) is (additively) generated by elements of the form [X]/L r with X smooth and proper and of pure dimension ≤ i + r. In arbitrary characteristic, F i K 0 (Var k ) is generated by elements of the form [X]/L r with X proper and of pure dimension ≤ i+r. To see this, note that by Noetherian induction, K 0 (Var k ) is generated by the classes of affine varieties. But affine varieties are in the subring generated by proper varieties, as they may be compactified, e.g. by considering their scheme-theoretic image under some quasi-projective embedding.
Unfortunately, it is difficult to understand the behavior ofK, as it is unknown whether or not L is a zero divisor. Thus, for much of this note, we will instead consider the completion R of K 0 (Var k ) at the ideal (L). That is, Remark 3. It is not known whether n (L n ) = (0).
That is, it is unclear whether the L-adic topology on K 0 (Var k ) is separated.
The completion R has three advantages: • Many well-known homomorphisms from K 0 (Var k ) to other rings (so-called "motivic measures" [11]) extend continuously to R, but not toK. For example, if k = F q , with q = p n , the homomorphism extends to a continuous homomorphism R → Z p . • R is easier to work with thanK-in particular, Theorem 12 below shows that K 0 (Var k )/L remembers exactly the stable birational geometry of smooth projective varieties, so convergence of limits in R "to first order" has geometric meaning.
• If k is algebraically closed of characteristic zero, we may define a surjection of topological rings Remark 18); if L is not a zero-divisor, then D is an isomorphism. For a smooth projective curve X, For a smooth projective surface S, Thus, we consider the following False Claim as an alternative to Conjecture 1-much of the work in this note will be aimed at examining the circumstances in which this claim fails. which we will disprove in Section 4, for k an algebraically closed field of characteristic 0. Before disproving these claims, however, we would like to take the odd step of motivating our False Claims 4 and 5, as well as the MSSP conjecture of Vakil and Wood. In particular, we will make a case that these false claims are natural, despite their falsehood.
First, let C be a smooth proper genus g curve over a field k, with a k-rational point. Then for n ≫ 0, Sym n C is a (Zariski) P n−g -bundle over Jac(C), and so [Sym n C] = (1 + L + · · · + L n−g )[Jac(C)] in K 0 (Var k ). Thus the limit lim clearly exists in R and equals (1) [Jac(C)](1 + L + L 2 + · · · ).
So False Claim 4 holds for smooth proper curves with a rational point. Similarly, False Claim 4 holds for connected rational or uniruled surfaces, and for connected varieties X whose classes [X] ∈ K 0 (Var k ) are polynomial in L, e.g. (split) affine algebraic groups and their homogeneous spaces. The proofs of these claims are not difficult, so we omit them. These examples also provide evidence for MSSP; indeed, every X for which MSSP is known to hold also satisfies False Claim 4, and vice versa.
MSSP and False Claim 4, when true, provide an algebro-geometric analogue of the following beautiful theorem of Dold and Thom.
Let us compare this to the situation in False Claim 4 for X a smooth proper curve of genus g over C.
In this case, Jac(X) has the homotopy type of a K(H 1 (X, Z), 1) by construction, and H 2 (X, Z) = Z, so K(H 2 (X, Z), 2) ≃ w CP ∞ ≃ lim − → CP n , which we may view as being represented by the class Likewise our computation (1) above gives in R. So we may view the Dold-Thom theorem as evidence for False Claim 4. Finally, consider the case k = F q , where q = p n . In this case, there is a "point-counting" ring homomorphism ψ q : K 0 (Var k ) → Z sending a variety X to #X(F q ). By analogy to the zeta function appearing in the Weil conjectures, Kapranov [8, 1.3] introduced the following "motivic zeta function," associated to a variety X over an arbitrary field: In the case that k is finite, is everywhere convergent as a power series over R. If X is a curve, the limit on the left specializes under ψ q to the "analytic class number formula" for the zeta functions appearing in the Weil conjectures; we have that (2) res t=1 ζ X (t) = # Jac(X)(F q ) 1 − q and likewise Thus we may view False Claim 4 or MSSP as analogues of analytic class number formulas. To put it another way, these claims are analogues of the fact that the zeta functions appearing in the Weil conjectures have a pole of order 1 at t = 1-assuming the power series expansion for (1 − t)Z mot X (t) is valid at t = 1. We will expand on this last heuristic in Section 5.
Remark 7. We refer to Equations (2) and (3) as "analytic class number formulas" because of their (informal) resemblance to the analytic class number formula for Dedekind zeta functions associated to number fields.
Remark 8. By analogy to the Weil conjectures, one might guess that Z mot X (t) is the power series associated to a rational function with coefficients in K 0 (Var k ). Kapranov shows that this is true for curves with a rational point [8, (1.3.5)(a)], where the hypothesis of the existence of a rational point is left implicit. For curves with no rational point the argument does not work. The issue is that the usual Picard functor is not representable in this case, and so Sym n (X) is not a projective space bundle over Pic n (X), which is an obstruction to Kapranov's argument. It is unclear to the author if this issue can be rectified. On the other hand, Larsen and Lunts have shown [11,12] that these zeta functions are not rational over K 0 (Var k ) for most surfaces. The question of the rationality of Z mot is open. The plan for the rest of this note is as follows. In Section 3 we will introduce several facts and conjectures about K 0 (Var k ) and discuss their interplay. In Section 4 we will disprove False Claims 4 and 5 and deduce the conditional falsity of MSSP in the case of smooth projective surfaces X with non-vanishing H 0 (X, ω X ). In Section 5 we will propose Hodge-theoretic heuristics for the failure of MSSP and give evidence for them in terms of the "Newton polygons" of Kapranov zeta functions Z mot X (t) of curves.

Preliminaries and Discussion
From here on, unless otherwise stated, k will be algebraically closed of characteristic zero.
In [1], Bittner gives the following useful presentation of K 0 (Var k )-the proof uses resolution of singularities and weak factorization of rational maps. We will use her description of K 0 (Var k ) and some of her constructions to relate MSSP to False Claims 4 and 5.
Theorem 9 (Bittner [1, Theorem 3.1]). K 0 (Var k ), for k algebraically closed and of characteristic zero, is generated by the classes of smooth proper k-varieties, subject only to the following relations: for X proper, Y a smooth closed subvariety of X, and E the exceptional divisor of the blowup Bl Y (X).
Bittner uses this presentation to construct a "duality map" D : , which we will use heavily.
Corollary 10 (Duality map [1,Corollary 3.4]). There exists a unique ring homomorphism D : for X smooth and proper.
Proof. By Theorem 9, D is uniquely determined by its value on smooth proper varieties; we must check that Dividing by L dim(X) gives the claim. That D is a ring homomorphism follows from the additivity of dimension.
We will need one further result on K 0 (Var k ), due to Larsen and Lunts [11]; it also follows from Bittner's presentation of K 0 (Var k ).
Definition 11 (Stable birationality). Let X and Y be two varieties; recall that X is stably birational to Y if X × A n is birational to Y × A m for some m, n. If k is separably closed, SB denotes the monoid of stable birational equivalence classes of smooth, connected, proper k-varieties under the operation of Cartesian product. For future reference, we will list here two conjectures about K 0 (Var k ); we will show that MSSP fails conditional on the truth of either of these conjectures.
Conjecture 13 (Cut-and-paste conjecture, Liu and Sebag [13]). Let X and Y be varieties over k. If [X] = [Y ] in K(Var k ), then there exist disjoint locally closed subvarieties X i of X, Y i of Y , such that This latter conjecture is a common assumption for those working with the Grothendieck ring of varieties, e.g. in [13,Remark 16], [10, 7.1].
We also record consequences of these conjectures which will be required later.
Proposition 15 (Stable birationality inK). Let X and Y be irreducible varieties over k. Then if the Cut-and-Paste Conjecture (Conjecture 13) holds for k, we have that Proof. The idea of the argument is to translate this equality into an equality in K 0 (Var k ), and then apply the cut-and-paste conjecture there. If , a i ∈ Z, and I is a finite index set. Equivalently, for some N ≫ 0, in K 0 (Var k ). Rearranging terms, we have for J ⊂ I, J := {i ∈ I|a i < 0} that Now, if the cut-and-paste conjecture holds, the equality above implies that we may write as disjoint unions of isomorphic locally closed subsets. That is, X ′ and Y ′ are equidecomposable. But X ′ , Y ′ have exactly one connected component of dimension N -respectively, A N −dim(X) × X and A N −dim(Y ) × Y -and all other connected components have dimension less than N , as n i > dim(S i ). So A N −dim(X) × X, A N −dim(Y ) × Y must be birational. Thus X and Y are stably birational.
We also have a similar result contingent on the truth of Conjecture 14.
Proposition 16. Suppose L is not a zero divisor in K 0 (Var k ) (Conjecture 14). Then , where X is smooth, connected, and proper. Then as desired. The latter statement follows analogously (indeed, D((L n )) equals F −n K 0 (Var k )[L −1 ] for all n ≥ 0).
If L is not a zero divisor, we may define an inverse map. By the previous observation, D ′ (F 0 K 0 (Var k )[L −1 ]) is exactly the image of K 0 (Var k ) in K 0 (Var k )[L −1 ] via the natural inclusion; as L is not a zero divisor, this image is isomorphic to K 0 (Var k ) itself. So D ′ induces a map g : F 0 K 0 (Var k )[L −1 ] → K 0 (Var k ). We need to check that F −1 K 0 (Var k ) maps to (L). But the verification proceeds as in the previous paragraph.
. If X is smooth and proper, this isomorphism sends [X]/L dim(X) to the stable birational equivalence class of X.
Proof. This follows immediately from Theorem 12 and Proposition 16. For a smooth projective surface S, where we use that Hilb n (S) is smooth and projective for S smooth and projective [5, p. 167].

Stable Birationality Of Symmetric Powers
The main geometric content of this section is the following: Theorem 19. Let X be a smooth connected projective surface with h 0 (X, ω X ) = 0. Let m be a non-negative integer. Then for all sufficiently large n, Sym n (X) is not stably birational to Sym m (X).
The idea of the proof is to produce a moving family of unirational subvarieties of Sym n (X) of high dimension, contradicting the following theorem of Mumford.
Theorem 20 (Mumford,[15], corollary on page 203). There exists a codimension-one subvariety W of Sym n (X) so that if Y ⊂ Sym n (X)\W consists entirely of rationally equivalent 0-cycles, then Y has dimension at most n.
Mumford's proof uses the so-called "symplectic argument" [21,Chapter 10]. The idea is to show that a connected subvariety of Sym n (X) consisting of rationally equivalent zero-cycles and containing a generic 0-cycle must lie tangent to an isotropic subspace of any two-form on Sym n (X). One can construct generically non-degenerate two-forms on the smooth locus of Sym n (X) given any non-zero two-form on X, giving an upper bound on the dimension of most such varieties.
Proof of Theorem 19. Without loss of generality n > m. Assume for the sake of contradiction that Sym n (X)× A l is birational to Sym m (X) × A 2n−2m+l for some l. Then there exists a variety U which may be embedded as a dense open subvariety of both Sym n (X) × A l and Sym m (X) × A 2n−2m+l .
A non-empty fiber of the projection map π m : U → Sym m (X) is a dense open subset of A 2n−2m+l , and is thus of dimension 2n − 2m + l. Choosing any k-point x in the image of the other projection map π n : U → Sym n (X) there exists some k-point y ∈ Sym m (X) so that x ∈ π n (π −1 m (y)). Choosing x ∈ Sym n (X) lying away from the subvariety W of Sym n (X) coming from Theorem 20, we choose y as above and let Y = π n (π −1 m (y)). As π −1 m (y) is an open subset of affine space, Y is unirational; furthermore Y has dimension 2n − 2m, as the non-empty fibers of π n have dimension l. As Y is unirational, points in it correspond to rationally equivalent 0-cycles. So for n > 2m, Y \ (Y ∩ W ) is a subvariety of Sym n (X) \ W consisting of rationally equivalent 0-cycles, of dimension larger than n, which contradicts Theorem 20.
Corollary 21 (False Claims 4 and 5 are false). Let X be as in Theorem 19. Then the classes [Sym n (X)] do not stabilize in K 0 (Var k )/L. Before proving this Corollary, we will need the following result of Göttsche. Here Hilb n (X) the Hilbert scheme of length n subschemes of X and P (n) is the set of partitions of n. If α ∈ P (n) we write α = (1 α1 , 2 α2 , · · · , n αn ), and define |α| = i α i .
Proof of Corollary 21. As Hilb n (X) is birational to Sym n (X) [5, p. 161] via the Hilbert-Chow morphism, we have from Theorem 19 that Hilb n (X) is not stably birational to Hilb m (X) for n ≫ m. Furthermore, as X is a smooth projective surface, Hilb n (X) is smooth and projective [5, p. 167], so [Hilb n (X)] maps to its stable birational equivalence class via the isomorphism sb : as Sym n (X) and Hilb n (X) are birational. But Hilb n (X) is smooth and proper, so its image in Z[SB] via the map defined in Proposition 16 and Corollary 17 is its stable birational equivalence class. These images do not stabilize, by Theorem 19.

Heuristics for Failure of MSSP, and Newton Polygons
In this final section, we will describe a heuristic predicting for which X False Claim 5 (and conditionally MSSP) will fail, in terms of the Hodge numbers of X; for X a curve, we will give some evidence for this claim.
For example, the 1-dimensional Hodge polygon of a smooth proper curve of genus g has slope 0 on [0, g] and slope 1 on [g, 2g].
If X is defined over a finite field F q and admits a lift X 0 to characteristic zero, there is a well-known relationship between the zeta function ζ X (t) appearing in the Weil conjectures, and the Hodge polygons of X 0 . Namely, ζ X (t) is an alternating product of polynomials p i (t), where p i (t) is the determinant of (I − Frob q t) acting on the i-th ℓ-adic cohomology group of X Fq with constant coefficients. Dwork [4], Mazur [14], and Ogus [17] have proven that the Newton polygon of p m (t) (with respect to the valuation at q) lies above the m-dimensional Hodge polygon of X 0 .
We suggest an imprecise analogy with motivic zeta functions, which explains the failure of the convergence of the symmetric powers of X. Before expanding on this analogy, we give a precise version for curves.
Throughout this section, we will use the fact that if X is a smooth projective curve with a rational point, Pic d (X) admits a stratification by locally closed subvarieties so that the natural map Sym d (X) → Pic d (X) (sending a divisor to its associated line bundle) is a projective space bundle over each stratum. Furthermore, the fiber over a point L is naturally identified with PΓ(X, L). If ] is a power series over K 0 (Var k ), we will use [t k ]f (t) to denote the coefficient of t k in f .
It is well known ([8, Theorem 1.1.9], [12,Theorem 3.7], [16,Theorem 7.33]), that if X is a smooth projective curve with a rational point, of degree 2g; let us compute its coefficients. We have that Here we take [Sym n (X)] = 0 for n < 0. In particular, where the last equality follows from the fact that ω X is the unique degree 2g − 2 line bundle L with h 0 (X, L) = g (from Serre Duality and Riemann-Roch), and all other line bundles L ′ of degree 2g − 2 satisfy h 0 (X, L ′ ) = g − 1. So is not in (L). For this we need to understand a bit about the stable birational geometry of Sym g−1 (X) and Sym g (X).
Lemma 25. Sym g−1 (X) and Sym g (X) are not stably birational to one another.
Proof. Sym g (X) is birational to Jac(X), so it suffices to show that Sym g−1 (X) is not stably birational to Jac(X). Suppose to the contrary that there is a rational, birational map Sym g−1 (X)× A n → Jac(X)× A n−1 . Then in particular some open subset U of A n maps injectively to Jac(X)×A n−1 ; but as Jac(X) is an Abelian variety, the image of this map under the projection to Jac(X) is trivial. Thus U is entirely contained in some fiber of the projection, so U maps injectively to A n−1 . But U has dimension n, so this is impossible.
Remark 26. This argument shows in general that if X, Y are varieties with dim(X) > dim(Y ), and no rational curves pass through a general point of X, then X and Y are not stably birational. In particular, for C a smooth proper curve of genus g, and n ≤ g, Sym n−1 (C) is not stably birational to Sym n (C) (no rational curves pass through a general point of Sym n (C) as it is birational to a subvariety of Jac(C)).
] − [Sym g−1 (X)] mod L. As Sym g−1 (X), Sym g (X) are smooth and projective, Lemma 25 gives that the image of this difference in Z[SB] via sb is non-zero, so we have the claim.
Corollary 28 (The Newton polygon of (1 − t)(1 − Lt)Z mot X (t) lies below the 1-dimensional Hodge polygon of X). For x ∈ K 0 (Var k ), define v L (x) ∈ Z ≥0 ∪ {∞} to be the greatest integer n so that x ∈ (L n ). Define the L-adic Newton polygon of a polynomial p(t) = a i t i ∈ K 0 (Var k )[t] to be the lower convex hull of the set of points (i, v L (a i )) (when there is no risk of confusion, we will drop the modifier " L-adic"). Then the Newton polygon of (1 − t)(1 − Lt)Z mot X (t) lies below the 1-dimensional Hodge polygon of X. Proof. Recall that the 1-dimensional Hodge polygon of X consists of the segments [(0, 0), (g, 0)] and [(g, 0), (2g, g)].
Thus it suffices to show that v L (1) = 0, v L ([t g ](1 − t)(1 − Lt)Z mot X (t)) = 0 and v L (L g ) = g. That v L (1) = 0 is simply the statement that L is not a unit; to see this, note that it suffices to produce any homomorphism out of K 0 (Var k ) sending L to a non-unit. The homomorphism K 0 (Var k ) → Z[t] sending a smooth proper variety to its Poincaré polynomial [13, p.331] suffices.
Corollary 27 is exactly the statement that v L ([t g ](1 − t)(1 − Lt)Z mot X (t)) = 0. Finally v L (L g ) is greater than or equal to g by definition. But the Poincaré polynomial of L g is t 2g ; thus L g is not in (L g+1 ) (which maps to (t 2g+2 ) via the Poincaré polynomial), as desired.
In fact, we claim that the Newton polygon of (1 − t)(1 − Lt)Z mot X (t) is actually equal to the 1-dimensional Hodge polygon of X. To see this, it suffices to show that v L ([t n ](1 − t)(1 − Lt)Z mot X (t)) ≥ n − g for n > g.
Lemma 29 (The Newton polygon of (1 − t)(1 − Lt)Z mot X (t) lies above the 1-dimensional Hodge polygon of X). For n > g, [t n ](1 − t)(1 − Lt)Z mot X (t) ∈ (L n−g ). Proof. Let x ∈ X be a rational point. Let J a,b,c ⊂ Pic n (X) be the locally closed subset consisting of those line bundles L with h 0 (L) = a, h 0 (L(−x)) = b, h 0 (L(−2x)) = c. Now By Riemann's inequality, a ≥ n − g + 1; b ≥ n − g; c ≥ n − g − 1; furthermore 0 ≤ a − b, b − c ≤ 1. Now if a = b = c, the corresponding term in the sum above vanishes. If a > b, (that is, a = b + 1) and b = c, An identical argument works for the cases a = b > c or a > b > c. Thus [t n ](1 − t)(1 − Lt)Z mot X (t) is a sum of terms in (L n−g ) and so is in (L n−g ) itself.
Remark 30. This argument works for curves with a rational point in arbitrary characteristic, and implies the classical result that the q-Newton polygons of the zeta function ζ X associated to a smooth projective curve X over a finite field F q lie above its associated Hodge polygons. In particular, the q-Newton polygon of ζ X (t) = ψ q (Z mot X (t)) lies above the L-adic Newton polygon of Z mot X (t) (because ψ q (L) = q), which lies above the Hodge polygon of X.
Unfortunately, Proposition 33 does not give examples in higher dimensions, as Corollaries 21 and 23 use the existence of a desingularization S of Sym n (X) which satisfies S = Sym n (X) mod L; the existence of such a desingularization in dimension greater than 2 appears to be an open question [12,Question 6.7]. Ulyanov [19,Theorem 2] constructs a compactification of the configuration space of distinct, labeled points in X upon which the symmetric group acts with Abelian stabilizers; he remarks that desingularizing this compactification should be doable via existing methods. So perhaps this question is tractable. Ulyanov's compactification is a modification of the well-known Fulton-MacPherson compactification [6].