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Rational cohomology of the moduli space of trigonal curves of genus 5

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Abstract

We compute the rational cohomology of the moduli space of trigonal curves of genus 5. We do this by considering their natural embedding in the first Hirzebruch surface and by using Gorinov–Vassiliev’s method.

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Notes

  1. The dimension of the vector space of polynomials defining plane quintics is \({\text {dim}}{\mathbf {C}}\left[ x_0,x_1,x_2\right] _5=21.\) A polynomial \(f\in {\mathbf {C}}\left[ x_0,x_1,x_2\right] _5\) is singular at a point \(P\in {\mathbf {P}}^2\) iff \(\frac{\partial f}{\partial x_i}(P)=0,\,i=0,1,2.\) Hence the dimension of the vector space of polynomials defining plane quintic with a fixed singularity is \(21-3=18\).

  2. For the factor \({\tilde{B}}({\mathbf {P}}^2\backslash \{P\},3)\) this follows by Lemmas 2.3 and 2.6. While for the second factor, this can be deduced by computing the Borel–Moore homology of \({\mathbf {P}}^1\backslash \{\text {3 points}\}\) in terms of \({\mathfrak {S}}_3\text{- }\) representations, that is \({\mathbb {S}}_3(1)\) in degree 2 and \({\mathbb {S}}_{2,1}\) in degree 1.

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Acknowledgements

I am sincerely grateful to my advisor, Orsola Tommasi, for guiding me throughout this work and for her precious help. I also thank the anonymous reviewer for the helpful comments and suggestions.

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Correspondence to Angelina Zheng.

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Trivial configurations

Trivial configurations

As we promised in the computation of the spectral sequences in Tables 2 and 3, we now consider the remaining configurations and prove that they have trivial twisted Borel–Moore homology.

1.1 Configurations (42)–(43)

Both these configurations are equivalent to the configurations of singularities of a plane quintic that is the union of a conic and a singular cubic. To be more precise, in the first type, the two curves meet each other at 6 distinct points and P is any of the points of intersection, while in the second one they intersect at the singular point of the cubic, that is P (Fig. 4).

Fig. 4
figure 4

Configurations of type (42)–(43)

Both configuration spaces can be fibered over the space of conics through P. If we denote the conic by \({\mathcal {C}},\) the fibers will be respectively equal to \(B({\mathcal {C}}\backslash \{P\},5)\) and \(B({\mathcal {C}}\backslash \{P\},4).\) They both have trivial twisted Borel–More homology by Lemma 2.1.

1.2 Configurations (44)–(45)–(46)

These configurations are all obtained by blowing up a singular point in the configuration of type 37 in [7], defined by the intersection points of two lines and a cubic curve in \({\mathbf {P}}^2\) having one singular point.

To be more precise, configurations of these types correspond to the blow up at P, where P has to be an ordinary double point. In the first P is defined as the point of intersection between a line and the cubic, in the second it is the point of intersection between the two lines, and finally it is the singular point of the cubic (Fig. 5). Note that, in the first two configuration spaces, the cubic need not to be irreducible: it can decompose into three concurrent lines or into the union of a conic and a line tangent to it. However, this cannot happen for configuration (46), otherwise P would not be an ordinary double point. The two reducible cases, with P not an ordinary double point, define configurations (59) and (55), respectively. Configuration (55) was already considered as configuration (K), while configuration (59) will be considered later.

Fig. 5
figure 5

Configurations of type (44)–(45)–(46)

Configuration spaces of type (44), (45), (46) can then all be fibered over the space parametrizing the two lines rs. The fibers are isomorphic to the quotient of \(B({\mathbf {C}}^*,2)\times B({\mathbf {C}},3)\), \(B({\mathbf {C}},3)\times B({\mathbf {C}},3)\) and \(B({\mathbf {C}},3)\times B({\mathbf {C}},3)\), respectively, by the involution given by exchanging the two lines. Since they have all trivial twisted Borel–Moore homology, the homology of the configuration spaces will be trivial as well.

1.3 Configuration (49)

Configurations of type (49) are obtained by the same plane curve considered for type (J), where P is defined as the point of intersection of a conic and a line (Fig. 6).

Fig. 6
figure 6

Configuration of type (49)

Then \(X_{(49)}\) can be fibered over \({\tilde{B}}({\mathbf {P}}^2\backslash \{P\},4)\ni \{A,B,C,D\}.\) Once these points are fixed, we notice that the conic \({\mathcal {C}}\) passing also through P is uniquely determined. Therefore the fiber Y is itself a fiber bundle over the space \(L\cong {\mathbf {P}}^1\backslash \{\text {5 points}\}\) of lines not passing through any of the points ABCD and not tangent to \({\mathcal {C}}.\)

The fiber \({\mathcal {Z}}\) is defined as the space of conics, passing through \(\{A,B,C,D\}\), not tangent to \(l\in L\) and different from \({\mathcal {C}}.\) The space \({\mathcal {Z}}\) is isomorphic to \({\mathbf {P}}^1\backslash \{0,1,\infty \}\cong {\mathbf {C}}\backslash \{0,1\}.\) Moreover, determining a conic in \({\mathcal {Z}}\) is equivalent to choosing a point in l that is different to PQ and the 2 points of tangency \(T_1,T_2\) in l. Thus, \({\bar{H}}_{\bullet }({\mathcal {Z}},\pm {\mathbf {Q}})\) is \({\mathbf {Q}}\) in degree 1 and 0 in all other degrees. Note also that, when moving l around A,  for instance, the points of tangency in l are swapped. Therefore \(\pi _1(L)\) acts on \({\bar{H}}_{1}({\mathcal {Z}},\pm {\mathbf {Q}})\) anti-invariantly and the Borel–Moore homology of the fiber Y is defined by that of L with non-trivial coefficient system:

$$\begin{aligned} {\bar{H}}_{\bullet }(L;{\bar{H}}_1({\mathcal {Z}}))={\mathbf {Q}}, \qquad \text {in degree }0. \end{aligned}$$

Finally, notice that we are considering a local system on L that changes its sign under the action of any loop in \({\mathbf {P}}^1\) around any point removed. Therefore any \(\gamma \in {\tilde{B}}({\mathbf {P}}^2\backslash \{P\},4)\) transposing a pair of points must act on the fiber as the multiplication by -1. This means that the local system induced by the fiber on \({\tilde{B}}({\mathbf {P}}^2\backslash \{P\},4)\) is \(\pm {\mathbf {Q}}\) and by Lemma 2.3 the twisted Borel–Moore homology of \(X_{(49)}\) will be trivial.

1.4 5.4 Configurations (50)–(51)

In all the following configuration spaces P has to be a triple point. More precisely, they are defined by blowing up the curves at P (Fig. 7).

Fig. 7
figure 7

Configurations of type (50)–(51)

We can fiber the configuration spaces over the space parametrizing the pairs of lines. The fiber spaces will be then isomorphic to a quotient of \(B({\mathbf {C}},2)\times B({\mathbf {C}},2)\) and \({\mathbf {C}}^*\times B({\mathbf {C}},3),\) respectively, and they both have trivial twisted Borel–Moore homology.

1.5 Configuration (52)

As above, P must be again a triple point. In particular, configurations of type (52) are defined by two distinct conics meeting at P and three additional points ABC,  and a line l through P,  not meeting any of ABC (Fig. 8).

Fig. 8
figure 8

Configuration of type (52)

Then, the configuration space can be fibered over the space \({\tilde{B}}({\mathbf {P}}^2\backslash \{P\},3)\times ({\mathbf {P}}^1\backslash \{\text {3 points}\})\ni \{(\{A,B,C\},l)\},\) parametrizing the intersection points between the two conics and the choices for the line l. Once we have fixed l,  two points on it will uniquely determine the two conics. Hence, the fiber space will be \(B({\mathbf {C}},2)\) whose Borel–Moore homology will be considered with constant coefficients. In fact, when we exchange the two conics we are actually exchanging 2 couples of points in the configuration space: the two points lying on the line, and the two points of intersection between the exceptional divisor and the strict transforms of the two conics. On the other hand, there is a natural action of \({\mathfrak {S}}_3\) on the base space. By noticing that both factors have no \({\mathfrak {S}}_3\text{- }\)anti-invariant classes in their homologiesFootnote 2, the total space will have trivial twisted Borel–Moore homology.

1.6 Configurations (53)–(54)

These configurations are obtained by blowing up a point of intersection between two lines and a point of intersection between a line and a conic, respectively, in the set of 9 distinct points in \({\mathbf {P}}^2\) defined as follows. Three points ABC are in general position, defined as the intersection points of three distinct lines. The other six points are defined as the intersection points between the three lines \({\overline{AB}},{\overline{BC}},{\overline{AC}}\) and a conic not tangential to the lines. This is configuration 39 in [7] (Fig. 9).

Fig. 9
figure 9

Configurations of type (53)–(54)

As in [7], we want to fiber both configuration spaces over the spaces parametrizing the points of intersection between the three lines. When we choose P as one of these points, e.g. A,  the total space will be fibered over \(B({\mathbf {P}}^2\backslash \{P\},2)\) instead of \(B({\mathbf {P}}^2,3).\) On the other hand, when P is the intersection point between the conic and a line, the configuration space is fibered over a quotient of \(B({\mathbf {P}}^2\backslash \{P\},2)\times {\mathbf {C}}^*\ni (\{B,C\},A).\) The fiber space, denoted by Y in [7], will be in both cases the same, i.e. a fiber bundle over \(B({\mathbf {C}}^*,2)\times B({\mathbf {C}}^*,2)\), the configuration space of 2 points on each of \({\overline{AB}},{\overline{BC}}\), excluding ABC,  with fiber isomorphic to \({\mathbf {C}}^*\) minus a point. Therefore it will have the same Borel–Moore homology, which is \({\mathbf {Q}}\) in degree 5, \({\mathbf {Q}}(1)^2\) in degree 6, \({\mathbf {Q}}(2)\) in degree 7 and zero in all other degrees. However we will have to consider the action of the fundamental group of the new base space that is either \(B({\mathbf {P}}^2\backslash \{P\},2)\), or it contains it as a factor of a product. The fundamental group will then be the restriction of the symmetric group \({\mathfrak {S}}_3\) to \(\{B,C\}\): \({\mathfrak {S}}_2.\) Thus, we only need to consider local systems of coefficients corresponding to the restrictions of the representations of \({\mathfrak {S}}_3\): trivial and sign representation will restrict respectively to trivial and sign representation on \({\mathfrak {S}}_2,\) while the 2-dimensional irreducible representation restricts to the direct sum of the trivial and sign representation. We have that \({\bar{P}}(B({\mathbf {P}}^2\backslash \{P\},2);{\mathbf {Q}})={\mathbf {L}}^{-4}t^8\) and \({\bar{P}}(B({\mathbf {P}}^2\backslash \{P\},2),\pm {\mathbf {Q}})={\mathbf {L}}^{-3}t^6,\) by Lemmas 2.3, 2.4 and 2.6. So we will get a similar \(E^2\text{- }\)term of the spectral sequence, with the only difference that the action of \({\mathfrak {S}}_2\) on \({\bar{H}}_6(Y;\pm {\mathbf {Q}})={\mathbf {Q}}(1)^2\) now must be reducible.

figure e

The differentials \(d^2_{8,5}:E^2_{8,5}\rightarrow E^2_{6,6},\) \(d^2_{8,6}:E^2_{8,6}\rightarrow E^2_{6,7}\) must be non-trivial because otherwise we would get non-trivial classes in the main spectral sequence whose corresponding cohomology is not divisible by that of \(GL(2,{\mathbf {C}}),\) contradicting section 3.1. Therefore, also these configuration spaces have trivial twisted Borel–Moore homology.

1.7 Configurations (56)–(57)

These configuration spaces are the configurations of singularities obtained by blowing up the curves at P (Fig. 10).

Fig. 10
figure 10

Configurations of type (56)–(57)

Fig. 11
figure 11

Configurations of type (58)–(59)

Consider first configurations of type (56). Similarly to configurations (53), (54), we fiber the space over the points of intersection between the lines, that is \({\tilde{B}}({\mathbf {P}}^2\backslash \{P\},2).\) These two points, together with P,  uniquely determine the three lines. The other 4 points left, together with P, will uniquely determine the conic. The fiber space is then isomorphic to the quotient of \({\mathbf {C}}^*\times {\mathbf {C}}^*\times B({\mathbf {C}}^*,2)\) by the involution exchanging the first two factors. Thus it has twisted Borel–Moore homology equal to \({\mathbf {Q}}\) in degree 4 and \({\mathbf {Q}}(1)\) in degree 5. The fundamental group of the base space acts by exchanging the two points, and thus induces an \({\mathfrak {S}}_2\text{- }\)action on the line not passing through P that is the one described in Lemma 2.2. Therefore, the Borel–Moore homology class in degree 5 must be anti-invariant under such an action, while the class in degree 4 will be invariant. By applying Lemmas 2.3, 2.4 and 2.6 we have that the second page of the spectral sequence must have the following form:

figure f

where the differential will be an isomorphism.

The second configuration space can be fibered over the space parametrizing the three lines through P. Since it suffices to fix 5 points on those lines to determine the conic, the fiber space will be a quotient of \(B({\mathbf {C}},2)\times B({\mathbf {C}},2)\times {\mathbf {C}},\) which has trivial twisted Borel–Moore homology.

1.8 Configurations (58)–(59)

These configuration spaces are both defined by 5 lines in the projective plane. The first is obtained by blowing up any of the singular points of 5 lines in general position, while the second one is obtained by blowing up the point of intersection of three concurrent lines, in a plane quintic defined by such three lines and two additional lines meeting at a point outside the other lines (Fig. 11).

We can fiber both configuration spaces over the set of lines meeting at P,  thus:

$$\begin{aligned} X_{(58)}\rightarrow B({\mathbf {P}}^1,2)\qquad \text {and}\qquad X_{(59)}\rightarrow B({\mathbf {P}}^1,3). \end{aligned}$$

The fiber spaces will then be the spaces of the remaining lines defining the configuration that are, respectively, \(B({\mathbf {C}}^2,3)\) and \(B({\mathbf {C}}^2,2),\) by duality. Both have trivial twisted Borel–Moore homology by Lemma 2.1.

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Zheng, A. Rational cohomology of the moduli space of trigonal curves of genus 5. manuscripta math. 169, 603–631 (2022). https://doi.org/10.1007/s00229-021-01347-x

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