Cohomology of the moduli space of non-hyperelliptic genus four curves
We compute the intersection Betti numbers of the GIT model of the moduli space of non-hyperelliptic Petri-general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett-Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the HKKN equivariantly perfect stratification, a partial resolution of singularities and the Decomposition Theorem.
Moduli spaces of curves and their geometrically meaningful compactifications are a central topic in algebraic geometry. In particular, one wants to understand the topology of these spaces. From that perspective, the purpose of this paper is to compute the intersection Betti numbers of the moduli space of non-hyperelliptic Petri-general curves of genus 4. The canonical model of such curves is a complete intersection of a smooth quadric and a cubic surface in projective space. This moduli space hence carries a natural compactification as GIT quotient:
Fedorchuk [Fed12] proved this space to be the final non-trivial log canonical model for the moduli space of stable genus 4 curves, namely
where is the boundary divisor. We refer to the results of Casalaina-Martin-Jensen-Laza (see [CMJL12] and [CMJL14]) for the description of the last steps of the Hassett-Keel program for log minimal models of , arising as VGIT quotients of the parameter space of complete intersections. At the other extreme of the Hassett-Keel program, the rational cohomology of was computed by Bergström-Tommasi in [BT07], and that of by Tommasi in [Tom05].
The strategy to compute the intersection Betti numbers of relies on a general procedure developed by Kirwan to calculate the cohomology of GIT quotients (see [Kir84], [Kir85], [Kir86]). The crucial step of that method consists of the construction of a partial desingularisation , known as Kirwan blow-up, having only finite quotient singularities, and the computation of its Hilbert-Poincaré polynomial. In our case, this space will be constructed by blowing-up the loci in corresponding to triple conics in , curves with two or two singularities, called D-curves, and curves with two singularities of type , called A-curves.
Examples of application of Kirwan’s method are the topological descriptions of the moduli space of points on the projective line ([MFK94, §8]), of surfaces of degree 2 ([KL89]) and of hypersurfaces in ([Kir89]), with explicit complete computations only in the case of plane curves up to degree , cubic and quartic surfaces. More recently, the procedure has been applied to compactifications of the moduli space of cubic threefolds ([CMGHL]). This paper deals with the first application to a moduli space of complete intersections.
Our result is summarised by the following:
The intersection Betti numbers of and the Betti numbers of the Kirwan blow-up are as follows:
while all the odd degree (intersection) Betti numbers vanish.
The structure of the paper reflects the steps of Kirwan’s machinery. In Section 2, after recalling the construction of as the GIT quotient , we compute the equivariant Hilbert-Poincaré polynomial of the semistable locus, by means of the Hesselink-Kempf-Kirwan-Ness (HKKN) stratification naturally associated to the linear action of on the parameter space . In Section 3, we explicitly construct the partial desingularisation , by blowing-up three -invariant loci in the boundary of , corresponding to strictly polystable curves. Section 4 is devoted to the computation of the rational Betti numbers of the Kirwan blow-up : here the error term coming from the modification process is divided into a main and an extra contribution. The intersection Betti numbers of are computed in Section 5, via the Decomposition Theorem (cf. [BBD82]) of the blow-down operations. We conclude with a geometric interpretation of some Betti numbers, via a description of the generating classes of curves in the GIT boundary.
Notation and conventions
We work over the field of complex numbers and all the cohomology and homology theories are taken with rational coefficients. The intersection cohomology will be always considered with respect to the middle perversity (see [KW06] for an excellent introduction). For any topological group , we will denote by the connected component of the identity in and by the finite group of connected components of . The universal classifying bundle of will be denoted by . If acts on a topological space , its equivariant cohomology (see [AB83]) will be defined to be . The Hilbert-Poincaré series is denoted by
and analogously for the intersection and equivariant cohomological theories. If is a finite group acting on a vector space , then will indicate the subspace of elements in fixed by .
I wish to thank my advisor Klaus Hulek who proposed me this problem, for many helpful discussions and suggestions. I am also grateful to Yano Casalaina-Martin, Radu Laza and Orsola Tommasi for useful conversations and correspondence and to all the authors of [CMGHL] for kindly sharing it with me.
2. Equivariant stratification and Hilbert-Poincaré series
A smooth non-hyperelliptic curve of genus 4 is realised by the canonical embedding as a complete intersection of a quadric and a cubic surface in the projective space . If the quadric is smooth, the curve is said to be Petri-general and thus defines a point in the complete linear system
of curves of bidegree on . Since every such curve admits a unique pair of systems, it follows that these curves are abstractly isomorphic as algebraic curves if and only if they lie in the same -orbit.
We consider the reductive group , which is only isogenus to , but has the advantage to define a linearisation of the hyperplane bundle of . We will work with this linearisation throughout all the results. The action of on is induced by the natural action of on via change of coordinates and the -extension interchanges the rulings of . Geometric Invariant Theory [MFK94] provides a good categorical projective quotient
whose cohomology we aim to compute. In particular, intersection cohomology satisfies Poincaré duality, allowing us to compute the Betti numbers up to dimension . However, we prefer to carry out the computations in all dimensions for the sake of completeness, and to report also the results mod for the sake of readability.
2.1. The HKKN stratification
From the results in [Kir84], the first step in Kirwan’s procedure is to consider the Hesselink-Kempf-Kirwan-Ness (HKKN) stratification of the parameter space, which, from a symplectic viewpoint, coincides to the Morse stratification for the norm-square of an associated moment map.
In general, let be a complex projective manifold, acted on by a complex reductive group , inducing a linearisation on the very ample line bundle . We pick a maximal compact subgroup , whose complexification gives , and a maximal torus , such that is a maximal compact torus of . Before describing the stratification, we need also to fix an inner product together with the associated norm on the dual Lie algebra , e.g. the Killing form, invariant under the adjoint action of .
[Kir84] In the above setting, there exists a natural stratification of
by -invariant locally closed subvarieties , indexed by a finite partially ordered set such that the minimal stratum is the semistable locus of the action and the closure of is contained in , where if and only if or .
We briefly sketch the construction of the strata appearing in the previous Theorem 2.1 (see [Kir84] for a detailed description). Let be the weights of the representation (a.k.a. the linearisation) of on and identify with via the invariant inner product. After choosing a positive Weyl chamber , an element belongs to the indexing set of the stratification if and only if is the closest point to the origin of the convex hull of some nonempty subset of . We define to be the linear section of
The stratum indexed by is then
The heart of Kirwan’s results in [Kir84] is the proof that the equivariant Betti numbers of the strata sum up to the cohomology of the whole space.
If we denote by the stabiliser of under the adjoint action of , the equivariant Hilbert-Poincaré series of each stratum is
where is the set of semistable points of with respect to an appropriate linearisation of the action of (cf. [Kir84, 8.11]).
2.2. Stratification for curves in
We now come back to our case and start computing the equivariant Hilbert-Poincaré series . Since is compact, its equivariant cohomology ring is the invariant part under the action of of , which splits into the tensor product (see [Kir84, 8.12]). Then
In fact , where has degree 4, and , with deg()=2. The extension acts by interchanging and , while it fixes the hyperplane class . Therefore the ring of invariants is generated by , and :
Since deg()=4 and deg()=8, we have:
According to Theorem 2.2, we need to subtract the contributions coming from the unstable strata. In our case, the indexing set of the stratification can be visualised by means of the following Figure 1, called Hilbert diagram.
There are 16 black nodes in this square, and each of these nodes represents a monomial in , for . This square is simply the diagram of weights of the representation of on with respect to the standard maximal torus in . Each of the nodes denotes a weight of this representation, namely
There is a nondegenerate inner product (the Killing form) defined in the Cartan subalgebra in . Using this inner product, we can identify the Lie algebra with its dual , and the above square can be thought of as lying in .
The Weyl group coincides with the dihedral group of all symmetries of the square. It operates on the Hilbert diagram as follows: the first two involutions are reflections along the axes, while the third one is along the principal diagonal. It is easy to see that the grey region is the portion of the square which lies inside a fixed positive Weyl chamber .
By definition, the indexing set consists of vectors such that lies in the closure of the positive Weyl chamber and is also the closest point to the origin of a convex hull spanned by a nonempty set of weights of the representation of on . In this situation, we may assume that such a convex hull is either a single weight or it is cut out by a line segment joining two weights, which will be denoted by (see Figure 1).
The codimension of each stratum is equal to (see [Kir89, 3.1])
where is the number of weights such that , i.e. the number of weights lying in the half-plane containing the origin and defined by . Moreover, let be the subgroup of elements in which preserve , then is a parabolic subgroup, whose Levi component is the stabiliser of under the adjoint action of .
All the contributions coming from the unstable strata are summarised in Table 1.
In the Table 1, the elements is a generator of , with automorphism , which is a double cover of the maximal torus . For every , the first column of Table 1 shows the weights contained in the segment orthogonal to the vector (see Figure 1): then via the correspondence (2) one can obtain an explicit geometrical interpretation of the curve contained in each unstable stratum. The terms appearing in the second, third and fourth columns are determined easily from the Hilbert diagram. The computations in the last column follow from applying Theorem 2.2 to the action of on , in order to compute the equivariant cohomology of each unstable stratum (see Remark 2.1). We shall discuss some of these cases below, the others can be treated in an analogous way.
For instance, when contains only two weights, is a projective line, and except in the second row the subgroup is the maximal torus . In these cases, the equivariant Hilbert-Poincaré series can easily be seen to be
while in the case of the second row the stabiliser is a double cover of the maximal torus and the cohomology of the corresponding stratum is
In the third row, the segment orthogonal to contains three weights, hence and by Theorem 2.2 the equivariant cohomological series of the correspondent stratum is
The computations of the last two rows are similar: the linear section is acted on by the group . The first factor is central and acts trivially on , while the action of the second factor can be identified with the action on the space of binary cubic forms by change of coordinates. This leads to
The -equivariant Hilbert-Poincaré series of the semistable locus is
3. Kirwan blow-up
3.1. General setting
In general the equivariant cohomology of the semistable locus does not coincide with the cohomology of the GIT quotient, unless in the case when all semistable points are actually stable. This is not the case for us. The solution is given by constructing a partial resolution of singularities , known as Kirwan blow-up [Kir85], for which the group acts with finite isotropy groups on the semistable points . We briefly describe how it is constructed.
We consider again the setting, as in Section 2.1, of a smooth projective manifold acted on by a reductive group . We also assume throughout the paper that the stable locus is non-empty. In order to produce the Kirwan blow-up, we need to study the GIT boundary and stratify it in terms of the isotropy groups of the associated semistable points. More precisely, let be a set of representatives for the conjugacy classes of connected components of stabilisers of strictly polystable points, i.e. semistable points with closed orbits, but infinite stabilisers. Let be the maximal dimension of the groups in , and let be the set of representatives for conjugacy classes of subgroups of dimension . For every , consider the fixed locus
Kirwan showed [Kir85, §5] that the subset
is a disjoint union of smooth -invariant closed subvarieties in . Now let be the blow-up of along and be the exceptional divisor.
Since the centre of the blow-up is invariant under , there is an induced action of on , linearised by a suitable ample line bundle. If is the very ample line bundle on linearised by , then there exists such that is very ample and admits a -linearisation (see [Kir85, 3.11]). After making this choice, the set of representatives for the conjugacy classes of connected components of isotropy groups of strictly polystable points in will be strictly contained in (see [Kir85, 6.1]). Moreover, the maximum among the dimensions of the reductive subgroups in is strictly less than . Now we restrict to the new semistable locus , so that we are ready to perform the same process as above again.
After at most steps, we obtain a finite sequence of modifications:
by iteratively restricting to the semistable locus and blowing-up smooth invariant centres (cf. [Kir85, 6.3]).
Therefore, in the last step, is equipped with a -linearised ample line bundle such that acts with finite stabilisers. In conclusion, we have the diagram
where the Kirwan blow-up , having at most finite quotient singularities, gives a partial desingularisation of , which in general has worse singularities.
3.2. Kirwan blow-up for curves in
Coming back to our case, we need to find the indexing set of the Kirwan blow-up and the corresponding spaces , for all . Firstly, we need a description of the semistability condition for non-hyperelliptic Petri-general curves of genus 4.
[Fed12, 2.2] A curve is unstable (i.e. non-semistable) for the action of on if and only if one of the following holds:
contains a double ruling;
contains a ruling and the residual curve intersects this ruling in a unique point that is also a singular point of .
The GIT boundary is described by the following
Unions of a smooth double conic and a conic that is nonsingular along the double conic. These form a one-dimensional family;
Unions of three conics meeting in two singularities. These form a two-dimensional family;
Unions of two lines of the same ruling, meeting the residual curve in two singularities.
Now one can compute the connected components of the identity in the stabilisers among all the four families of polystable curves listed above, in order to find the indexing set . Nevertheless, we provide a more explicit, but equivalent, way to find the set , which has also the advantage to compute and .
We must find which nontrivial connected reductive subgroups fix at least one semistable point. Firstly, since is connected, must be contained in . Secondly, since we are interested only in the conjugacy class of , we may assume that its intersection with the maximal torus is a maximal torus of and is a maximal compact subgroup. Since is not a weight, it follows that fixes no semistable points. Therefore is a subtorus of rank one.
The fixed point set in consists of all semistable points whose representatives in are fixed by the linear action of . Thus is spanned by those weight vectors which lie on a line through the centre of the Hilbert diagram and orthogonal to the Lie subalgebra . Up to the action of a suitable element of the Weyl group , we can assume that the line passes through the chosen closed positive Weyl chamber . We have only two possibilities, see Figure 1.
Therefore we proved the following
If is a subgroup in the indexing set of Kirwan’s partial resolution, let denote the maximal torus of and let denote the fixed-point set of in . Then, up to conjugation, there are two possibilities for and :
and is contained in the projective space spanned by the polynomials , , and .
and is contained in the projective space spanned by the polynomials and .
We start analysing the second case. We can easily see from the characterisation of semistable points (Theorem 3.2) that all the semistable curves are given by with and . Geometrically these curves contains two lines of the same ruling and the residual curve intersects them in 2 points, giving 2 singularities of type . We will call these curves as A-curves. Their singular points are and in ; see the Figure 2.
By rescaling the variables and , all the semistable A-curves are equivalent to the curve defined by
Through this geometric description, it is now easy to show that in this case actually . We recall that is the connected component of the identity in the stabiliser of the A-curves: up to conjugation, we can think just of . Yet every element of , stabilising the point corresponding to in , will induce an automorphism of , which a fortiori must preserve the singular locus. Therefore every element of must fix and or interchange them. Hence
From the connectedness of , it follows , hence .
Now we analyse the first case. We can easily see via the Hilbert-Mumford numerical criterion ([MFK94, 2.1]) that all the semistable curves are given by
where are not simultaneously zero and are not simultaneously zero, i.e. . Moreover we can write every curve
as the union of three conics in the class , all meeting at points and in . We find three cases depending on how many ’s coincide.
Assume that all the coincide, namely the curve is a triple conic, which turns out to be equivalent to , defined by
This curve is nothing but a triple line diagonally embedded. Thus its stabiliser in is diagonally embedded, too. We get a non-splitting central extension of groups:
where is the stabiliser of in , that is to say the preimage of under the natural homomorphism . Here must be thought as the subgroup . Therefore we find the indexing subgroup diagonally embedded in and the associated spaces are one point.
Assume that two coincide and the third one does not. The semistable curves of this type are unions of a smooth double conic and a conic that is nonsingular along the double conic. They intersect at the points and , which consist of singularities of type ; see Figure 3.
Now we can argue like in the case of , noticing that every element of must preserve the singular points. Therefore , so that .
Assume all the are distinct from each other. The semistable curves of this kind are unions of three conics meeting in two singularities. These singular points are again and ; see Figure 4.
Arguing once more as before, we find that .
In conclusion, we proved the following:
The indexing set of the Kirwan blow-up, such as the fixed loci , for curves in , can be described as follows:
, diagonally embedded in , and in this case is the triple conic.
and in this case is the set of D-curves.
and in this case is the set of A-curves.
Moreover, the following holds:
We recall that the Kirwan’s partial desingularization process consists of successively blowing-up along the (strict transforms of the) loci in order of dim, to obtain the space , and then taking the induced GIT quotient with respect to a suitable linearisation. In our situation, we get the diagram