Moduli of curves on toric varieties and their stable cohomology Oishee Banerjee

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Moduli of curves on toric varieties

and their stable cohomology

Oishee Banerjee

Abstract

We prove that the cohomology of the moduli space of morphisms of a ﬁxed

ﬁnite degree from a smooth projective curve Cof genus gto a complete sim-

plicial toric variety P(Σ), denoted by the rational polyhedral fan Σ, stabilizes.

As an arithmetic consequence we obtain a resolution of the Batyrev-Manin

conjecture for toric varieties over global function ﬁelds in all but ﬁnitely many

characteristics.

1 Introduction

Noting that the moduli space of degree dmorphisms from a smooth projective

curve of genus gto the projective space Pn(or more generally, to weighted por-

jective stacks, see [BPS]and the references therein) is well studied so far as (étale

) homological stability is concerned, the next level of generalization is to ask the

same question for toric varieties- a question whose arithmetic counterpart is the

Batyrev-Manin type conjecture over global function ﬁelds.

To elaborate, ﬁx an algebraically closed ﬁeld K. Let Σbe a complete, simplicial

rational polyhedral fan in N⊗R∼

=Rr,Na lattice, and let Mbe the dual lattice

of N, their respective basis given by e1, . . . , er, and its dual e1, . . . , er. Let PΣbe

the corresponding complete simplicial toric variety, and Σ(1)denote the set of

rays, Σ(1):={ρ1, . . . , ρn}, and v1, . . . , vnthe integer generators of those rays. We

denote cones in Σby σ, the set of all rays in σby σ(1)and let Σmax denote the

set of maximal cones in Σ. A morphism from a smooth projective Cof genus gto

P(Σ)corresponds to what we call a non-degenerate Σ-collection which consists of

the following data:

i line bundles Lifor each ρi∈Σ(1),

ii global sections si∈H0(C,Li)for each ρi∈Σ(1),

iii compatibility isomorphisms: αei:NρL〈ei,vj〉→ OC,

iv satisfying the nondegeneracy condition: suppose s∗

j:L−1

j→ OCdenotes the map

induced by the section sjfor each ρj∈Σ(1), then there is a surjection of locally

free sheaves M

σ∈Σmax ⊗ρ /∈σ(1)s∗

ρ:M

σ∈Σmax ⊗ρ /∈σ(1)L−1

ρ→ OC.

arXiv:2210.05826v1 [math.AG] 11 Oct 2022

CURVES ON TORIC VARIETIES

If the last condition of non-degeneracy is not satisﬁed we simply call it a Σ-

collection.

Remark 1.1. Note our distinction in terminology from what Cox uses- Σ-collections

in [Cox1]are, by deﬁnition, non-degenerate.

In the language of Cox (see [Cox1]) the collection of triples (Lρ,sρ,αu)ρ∈Σ(1),u∈M

satisfying the above conditions is a Σ-collection on C(note that the deﬁnition of

Σ-collections holds for any K-scheme X, not just for a smooth projective curve).

Let ~

d:= (d1, . . . , dn)∈Nnbe a degree vector, where n:=|Σ(1)|and let Mor~

d(C,P(Σ))

denote the moduli space of degree ~

dmaps from Cto P(Σ)given by Σ-collections

on Csuch that degree of Li=di(note that by condition iii above, this forces

Pidiui=0 for the space of morphisms to be non-empty). A primitive collection

(see [CLS]) of rays in Σis a subset of Σ(1)which does not span a cone in Σ, but

every subset of that collection does. E1, . . . , Etbe its primitive collections. We now

state our main theorem.

Theorem 1. Let C be a smooth projective curve of genus g, P(Σ)a complete sim-

plicial toric variety deﬁned by a rational polyhedral fan Σ, let {ρ1, . . . , ρn}be its

rays, and let ~

d:= (d1, . . . , dn)∈Nnbe a degree vector such that di≥2g for all i and

Pdiui=0. Let n0:=min{d1, . . . , dn}−2g. Then there exists a second quadrant spec-

tral sequence, which converges to H∗(Mor~

d(C,P(Σ));Q)an algebra, which has the

following description. The E2term is a bigraded algebra that collapses on E−p,q

2p≤n0

Furthermore, E−p,q

2p≤n0

is a quotient of the graded commutative Q-algebra

H∗(J(C);Q)⊗(n−r)⊗Q[D1, . . . , Dn]/I⊗⊕t

j=1∧Q{zj}⊗⊕t

j=1SymQ{αj

1, . . . , αj

2g},

where t is the number of primitive collections of Σ, Hi(J(C);Q)has degree (0, i), Di

has degree (0, 2)for all i, zjhas degree (−1, 2εj),αj

ihas degree (−1, 2εj−1)for all

i, modulo elements of degree (−i,j)with i >n0and the ideal I is generated by all

1. Di1·. . . ·Dikfor vi1, . . . , viknot in a cone of Σ;

2. Pr

i=1〈v,ui〉for v ∈M;

3. Pk

j=1Di1. . . c

Dij. . . Dikfor every primitive collection {ui1, . . . , uik} ⊂ Σ(1), where

Dijdenotes the jth entry removed.

Furthermore this is a spectral sequence of mixed Hodge structures, with Q{αj

1, . . . , αj

2g}

carrying a pure Hodge structure of weight 2εj−1and Q{zj}carrying a pure Hodge

structure of weight 2εjfor all 1≤j≤t, and Diis of type (1, 1)for all i.

In the case when C=P1the result above simpliﬁes exponentially, in a sense,

thanks to the class group of P1being trivial.

OISHEE BANERJEE

Theorem 2. Let PΣbe a complete simplicial toric variety deﬁned by the rational

polyhedral fan Σand let {ρ1, . . . , ρn}be its rays. Let ~

d be a degree vector satisfying

Pdiui=0. Then

Mor~

d(P1,P(Σ)) ∼

=Q[D1, . . . , Dn]/I⊗⊕1≤j≤t∧Q{zj}

where t is the number of primitive collections of Σ, zjhas cohomological degree 2εj−1

and of type (εj,εj), Diof cohomological degree 2and type (1, 1)and I is the ideal

generated by all

1. Di1·. . . ·Dikfor vi1, . . . , viknot in a cone of Σ;

2. Pr

i=1〈v,ui〉for v ∈M;

3. Pk

j=1Di1. . . c

Dij. . . Dikfor every primitive collection {ρi1, . . . , ρik} ⊂ Σ(1), where

Dijdenotes the jth entry removed.

An immediate consequence of applying the Grothendieck-Lefschtez ﬁxed point

theorem (see [Grothendieck]) to the cohomology results above is that the Fq-points

of these moduli spaces is of the order of qPn

i=1di−ng+r, where dim Mor~

d(C,P(Σ)) =

i=1di−ng +r.

Some context.

1. On toric varieties. The moduli space of curves in toric varieties, under var-

ious constraints has been an object of interest from the angles of tropical

geometry and enumerative geometry for a while now. The moduli space of

stable maps naturally occur in enumerative geometry and mirror symmetry.

However this note is entirely unrelated to those. Our focus is not stable, but

actual algebraic maps from smooth projective curves to complete simplicial

toric varieties- their topology and arithmetic, and thus working towards an

entirely different goal than enumerative geometry.

2. On morphism-spaces and Batyrev-Manin type conjectures. Morphism-spaces

have been studied in various guises from various angles.

• In topology, there is an inﬂuential paper of Segal ([Segal])where he

worked speciﬁcally with projective spaces, using scanning maps to com-

pute the stable homology of what he calls the moduli space of ‘based’

holomorphic maps Rat∗

n(CP1,CPr), the moduli space of (r+1)polyno-

mials in one variable of degree n, up to C×. He generalizes it to higher

genera in the domain. To this date, to the best of our knowledge, all

generalizations (e.g replacing P1by Pnwith n≥1, or replacing CPrby

a Grassmanian) all critically uses the key idea of Segal’s. From the point

of view topology, the last known activity seems to be Guest’s beautiful

paper (see [Guest]) where he shows that the moduli space of holomor-

phic maps of a ﬁxed degree from P1to P(Σ)(which, he assumed is

CURVES ON TORIC VARIETIES

smooth and projective), has relations with conﬁguration spaces of C

with labels in a partial monoid, and uses this connection to show that

this moduli space is homotopy equivalent, up to a certain range (which

he computed in the case of some speciﬁc examples of P(Σ)) i.e. stably.

• In number theory, in a landmark paper by Browning and Sawin (see[BS])

they handle the case when the target projective space is replaced by a

hypersurface of low enough degree by using a geometric version of the

Hardly-Littlewood circle method.

In a sense lying in the middle, our methods are, on one hand, homotopy

theoretic resulting in the understanding of the topology of Mor~

d(C,P(Σ)),

and on the other hand all maps are algebraic, thus keeping track of the arith-

metic data every step of the way. In turn we see a much stronger version

of Guest’s with techniques that bypass Segal’s scanning maps. To be pre-

cise, our computation in the case of P1is the complete cohomology ring (i.e.

not just the stable part, unlike Guest’s) with Mixed Hodge structure, and for

higher genus, a spectral sequence which gives the stable cohomology when

the degree is large enough.

There’s one slight caveat– unlike Guest, we work with cohomology with ra-

tional coefﬁcients (whereas his are with integer coefﬁcients). Although, the-

oretically, our methods can be moulded to work for cohomology with Z-

coefﬁcients in practice the computations get unnecessarily difﬁcult, with no

impact to the arithmetic implications.

Plan of attack. Keeping aside technical algebro-geometric difﬁculties stemming

from genus considerations, the method for both the theorems is essentially ho-

motopy theoretic. It makes crucial use of [Banerjee21], or more accurately the

proof of it, and is, in some ways, naturally similar to the case when the toric va-

riety in question is replaced by a weighted projective stack (whose generic point

has a non-trivial stabilizer)- a problem answered by Banerjee, Park and Schmitt in

[BPS]. This also brings to the forefront the fact that our method behaves well in

the stacky world because the key lemma in our proof ([Banerjee21, Lemma 2.11])

is a statement that holds with sheaves of Q-vector spaces as well as `-adic sheaves

in étale topology. To portray the general philosophy behind these results we ﬁrst

prove the case of C=P1and then move on to the higher genus case.

2 Cox ring and its generalization to toric bundles.

In this section we brieﬂy recall Cox’s method (see [Cox2]) of attaching a homoge-

nous polynomial ring to a toric variety P(Σ)(which need not be complete, or sim-

plicial for most deﬁnitions to work, but we still assume them for our purposes), a

monomial ideal, and a reductive linear algebraic group, such that P(Σ)can be con-

structed as a categorical quotient (which is a geometric quotient in case the toric

variety is complete and simplicial), much like we attach a homogenous coordinate

ring, the irrelevant ideal to a projective space and the 1-dimensional torus Gm.

OISHEE BANERJEE

Then we use it in a relative setting, deﬁning what is called a toric bundle, and

we compute its cohomology (which we use in the proof of Theorem 1.)

Very brieﬂy we recall parts of [Cox2]. Let Σbe a rational polyhedral fan, and

let P(Σ)be the corresponding toric variety. Let Σ(1) = {ρ1, . . . , ρn}be the set of

rays. The homogenous coordinate ring of P(Σ)is deﬁned by

K[x1, . . . , xn]

(with grading given by the class group Cl(Σ)of P(Σ), which we will need later).

The exceptional locus Z(Σ)is deﬁned as the space in An=Spec(K[x1, . . . , xn]) cut

out by the monomial ideal B(Σ)⊂K[x1, . . . , xn]which is generated by monomials

of the form

xi1·. . . xiεk

for each primitive collection Ek={ρi1, . . . , ρεk}of rays. Let G(Σ):=Hom(Cl(Σ),Gm)

act on An−Z(Σ)by

g·(t1, . . . , tn):= (g(D1)t1, . . . , g(Dn)tn)

and the resulting geometric quotient (see [CLS]or [Cox2, Theorem 2.1]) is P(Σ).

Mimicking the case of projective spaces, we will denote a point in P(Σ)by [t1:

. . . : tn]when there is no confusion, which denotes an equivalence class of a point

(t1, . . . , tn)∈An−Z(Σ)under G(Σ).

One can, of course, make a similar construction in a relative setting i.e. over a

base scheme X. Let Fbe a vector bundle on Xthat splits into sub-vector bundles

F1⊕. . . ⊕ Fn. The action of the reductive algebraic group G(Σ)on Fis locally

given by

g·(s1, . . . , sn):= (g(D1)s1, . . . , g(Dn)sn)

where for all sections siof Fi, 1 ≤i≤n. The exceptional locus ZX(Σ)is con-

structed likewise and, by abuse of notation, denoting the total space of Fby F,

we call the geometric quotient

F − ZX(Σ)/G(Σ)

atoric bundle, with ﬁbres naturally isomorphic to P(Σ), and denote it by PX(F,Σ).

When Xis a point and in turn, Fan afﬁne space, we simply write (as we already

did) P(Σ).

The cohomology of H∗(P(Σ)) is well known (see [CLS]or [Fulton]):

Proposition 2.1. For a complete simplicial toric variety P(Σ),

H∗(P(Σ);Q)∼

=Q[D1, . . . , Dn]/I

where Dihas cohomological degree 2and type (1, 1)and I is the ideal generated by

all

1. Di1·. . . ·Dikfor {ρi1, . . . , ρik}a primitive collection in Σ(1);

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Moduli of curves on toric varieties and their stable cohomology Oishee Banerjee

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