The minimality of biharmonic hypersurfaces in pseudo-Euclidean spaces

1.   Introduction

Let $X$ be a connected projective smooth curve over an algebraically closed field $k$ and $G\subseteq \operatorname{Aut}(X)$ be a finite subgroup. Then $G$ acts in a natural way on the space of the holomorphic differential forms on $X$, and thus we obtain a $k$-linear representation $G \rightarrow \operatorname{GL}\left(H^{0}(X, \Omega_{X})\right)$. In other words, $H^{0}(X, \Omega_{X})$ is a $k[G]$-module.

If we want to study the $k[G]$-module structure on $H^{0}(X, \Omega_{X})$, a basic problem is to determine the multiplicity of each indecomposable representation on it. The group algebra $k[G]$ is reductive (or we say semisimple) if and only if the characteristic $\operatorname{char}(k)$ is either zero or coprime with the order of $G$; in this case the indecomposable representations of $G$ are exactly its irreducible representations.

The Chevalley-Weil formula tells us that the multiplicity of each irreducible representation can be characterized by the genus of the quotient curve $X/G$ and the ramification data in the quotient map $\pi:X\rightarrow X/G$, which is a branched (or ramified) cover.

1.1.   Origins and development of the Chevalley-Weil formula

The study of the Chevalley-Weil formula can be traced back to the late 19th century when Hurwitz ^[8] studied finite cyclic automorphism groups on compact Riemann surfaces ($k = \mathbb{C}$). Later, in the 1930s, Chevalley and Weil ^[2] calculated the irreducible multiplicities for a general finite group $G$ in the unramified case of $\pi$. Shortly after, Weil independently resolved the ramified case ^[18]. This result is therefore named the Chevalley-Weil formula, and the formula also holds for the characteristic $p$ case with $k[G]$ semisimple^[9].

When $\operatorname{char}(k) = p\mid \#G$, the structure of $H^{0}(X, \Omega_{X})$ becomes more complicated, so people usually impose some restrictions on the cover $\pi$ and the group $G$. There has been some research on the case where the ramified cover is tamely ramified ^[9,16] or weakly ramified ^[10]. In addition, there has been research on some special automorphism groups, such as cyclic groups ^[17], finite $p$-groups ^[5], or groups with cyclic Sylow subgroups ^[1]. There are also some studies over perfect fields ^[12].

For general theory, in 1980, Ellingsrud and Lønsted ^[4] used the language of equivariant K-theory to study the Lefschetz trace of coherent $G$-sheaves on projective varieties for $k[G]$ semisimple. They obtained precise results of the $k[G]$-module structure on $H^{0}(X, \mathscr{L})$ where $\mathscr{L}$ is an invertible $G$-sheaf and $X$ is a smooth projective curve [4, Theorem 3.8], which is a generalization of the Chevalley-Weil formula.

For higher dimensions, Nakajima gave some basic results in the 1980s. Consider a finite étale Galois cover $f:X\rightarrow Y$ between two projective algebraic varieties over a field $k$ with $G = {\rm{Gal}}(X/Y)$. Then for any coherent $G$-sheaf $\mathcal{F}$, there exists a finitely generated free $k[G]$-module complex

such that $H^{i}(X, \mathcal{F})\simeq H^{i}(L^{\bullet})$ as $k[G]$-modules^[15]. The approach basically follows Mumford's method in [14, $II.5$ Lemma 1].

When the étale condition on the cover $f$ is replaced by the requirement that $f$ be tamely ramified, the result will be weaker: the above finitely generated free $k[G]$-module complex is just a finitely generated projective $k[G]$-module complex ^[16]. It leads to a corollary that $H^{n}(X, \mathcal{F})$ is $k[G]$-projective when $H^{i}(X, \mathcal{F}) = 0$ for all other $i\neq n$. Now for a non-empty finite $G$-invariant set $S$ in the curve $X$, we have $H^{1}(X, \Omega_{X}(S)) = 0$ (and therefore $H^i (X, \Omega_X (S)) = 0$ for all $i > 0$). As a direct consequence of the above result, when $\pi$ is tamely ramified, the logarithmic differential space $H^{0}(X, \Omega_{X}(S))$ is a projective $k[G]$-module. It should be noted that the field above need not be algebraically closed.

Almost at the same time, Kani described $H^{0}(X, \Omega_{X})$ through the study of logarithmic differential space $H^{0}(X, \Omega_{X}(S))$, also obtaining that $H^{0}(X, \Omega_{X}(S))$ is projective ^[9]. However, at the end of his paper, Kani also pointed out that most of his results were covered by Nakajima's work. Nevertheless, Kani's proof process is more precise and also provides valuable tools for the main results of this paper.

For smooth curves, the Chevalley-Weil formula was well understood by now, but no attempt has been made for singular curves. The case of a nodal curve is a good frame in which to generalize the Chevalley-Weil formula.

1.2.   Key findings and goals

Now let $X$ be a connected projective nodal curve over an algebraically closed field $k$, $G\subseteq \operatorname{Aut}(X)$ is a finite subgroup of order $n$ with $k[G]$ semisimple, and $\pi:X\rightarrow Y = X/G$ is the quotient map. Then $Y$ is also a nodal curve.

In Section 2.1, we demonstrate that the regular differentials of nodal curves are suitable generalizations of the holomorphic differentials. A regular differential $\varphi\in H^0(X, \omega_X)$ is essentially a log differential in $H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X))$ that satisfies certain residue relations, where $\hat{S}_X$ is the preimage of the singular points under the normalization $\hat{X}\rightarrow X$. Therefore $G$ acts (right) naturally on the regular differential space $H^0(X, \omega_X)$. Here $\omega_X$ is the dualizing sheaf of $X$.

In this paper, we set the goal to calculate the multiplicity of every character $\chi:G\rightarrow k^{\times}$, that is, the dimension of the eigenspace:

Remark 1.1. Every $1$-dimensional representation is its own character. Although characters are the simplest irreducible representations, it still contains a lot of information that we are concerned about. There are mainly two reasons why we study it:

● When $G$ is cyclic (or abelian), all irreducible representations of $G$ are 1-dimensional;

● The space of $G$-invariant regular differentials is just the eigenspace of trivial character $1_G:G\rightarrow k^{\times}$, namely $H^{0}(X, \omega_{X})^G = H^{0}(X, \omega_{X})_{1_G}$.

Recall that when $X$ is smooth, every $G$-invariant holomorphic differential on $X$ is the pullback of a holomorphic differential on the quotient curve $Y = X/G$ (see Proposition 3.2). Consequently,

where $g_Y$ denotes the (geometric) genus of $Y$.

When $X$ is a nodal curve, we establish that

proven for the irreducible case in Proposition 3.5 and extended to the general case (where $X$ has multiple irreducible components) in Proposition 4.3. This leads to

where $p_a(Y)$ is the arithmetic genus of $Y$. This result generalizes the smooth case mentioned earlier.

For an arbitrary character $\chi: G \rightarrow k^{\times}$, when $X$ is smooth, we have:

In this expression:

● $g_Y$ is the genus of $Y$.

● The term $m_{\chi}$ is defined as

Here:

– The sum runs over all points $Q \in Y$.

– $e_Q: = e_P$ is the ramification index at any $P\in \pi^{-1}(Q)$. Note that $e_Q > 1$ if and only if $Q$ is a branch point, with $R_{G, Q} = 0$ at all other points; therefore the sum is finite.

● $R_G$ denotes the ramification module of $\pi$ and $R_{G, Q}$ is the ramification module at the point $Q$ (see Section 2.4).

● The notation $\langle\chi, V\rangle_G: = \operatorname{dim}_k \operatorname{Hom}_{k[G]}(W, V)$ represents the multiplicity of $\chi$ in a $G$-representation $V$, where $W$ is the irreducible $k[G]$-module corresponding to $\chi$.

● $\lfloor a \rfloor$ denotes the greatest integer less than or equal to $a$.

When $X$ is a nodal curve, we require the quotient curve $Y = X/G$ to be smooth^* (see Remark 3.9).

^*Note that the smoothness assumption is imposed for a general character $\chi$, while in the case of the trivial character $\chi = 1_G$, the quotient $Y = X/G$ is permitted to be nodal.

For the irreducible case, we have

where $\hat{X} \rightarrow X$ is the normalization. We present the Chevalley-Weil formula for irreducible nodal curves in Theorem 3.13:

In this expression:

● $\hat{S}_X^{\chi}\subset \hat{S}_X$ is the singular $\chi$-set of $\pi$ (see Proposition 3.7).

● The term $m_{\chi}(\hat{S}_X^{\chi})$ is an integer defined as

Here:

– $\hat{\pi}: \hat{X} \rightarrow Y$ is the projection map after normalization.

– $\#\hat{\pi}(\hat{S}_X^{\chi})$ denotes the number of points in the image of $\hat{S}_X^{\chi}$ under $\hat{\pi}$.

– $e_Q$ and $R_{G}, R_{G, Q}$ are defined as before, but for $\hat{\pi}$ this time.

Remark 1.2. Compared to the smooth case, the difference in $\operatorname{dim}_k H^{0}(X, \omega_{X})_{\chi}$ arises between $m_{\chi}$ and $m_{\chi}(\hat{S}_X^{\chi})$. The latter incorporates additional ramification data at singularities (the singular $\chi$-set). Note that when $\hat{S}_X^{\chi} = \varnothing$, we have

and $m_{\chi}(\hat{S}_X^{\chi}) = m_{\chi}$, just as in the smooth case.

In particular, when $X$ is smooth, we have $\hat{X} = X$, so this formula is a direct generalization of the smooth one.

In the general case, let $X = \cup^{d}_{i = 1} X_i$ be the decomposition into irreducible components. We can reduce the computation to the irreducible case.

Let $\alpha_1: \hat{X}_1 \rightarrow X_1$ be the normalization of $X_1$, and then $\hat{X}_1 \overset{\hat{\pi}_1}{\rightarrow } Y\overset{\sim}{ = }X_1/G_1$ is the induced covering map of smooth curves and we have

Here:

● $G_1: = \{\sigma\in G\mid \sigma(X_1) = X_1\}$ is the stabilizer subgroup of $G$ on the component $X_1$.

● $\chi_1: G_1 \rightarrow k^{\times}$ is the restriction of $\chi$ to $G_1$.

● $\hat{S}_{X_1}^{\chi_1}$ is the singular $\chi_1$-set of $\hat{\pi}_1$.

● $I_1$ denotes the intersection locus in $X$ (comprising all intersection points of the irreducible components) restricted to $X_1$ and

Here $G_{P}$ is the stabilizer subgroup.

● $\hat{S}_{X_1}^{\chi_1} \cap I_1^{\chi} = \varnothing$, since the points in $I_1$ are all nodes in $X$, but smooth points in $X_1$.

By applying the same argument as in the irreducible case, we obtain the Chevalley-Weil formula on connected nodal curves in Theorem 4.5:

● See the term $m_{\chi_1}(\hat{S}_{X_1}^{\chi_1}\sqcup I_1^{\chi})$ in (4.14).

● The final term $\delta_{\chi} = 0$ or $1$ where $\delta_{\chi} = 1$ if and only if $I_1^{\chi} = \varnothing$ and $\chi_1 = 1_{G_1}$.

Remark 1.3. Compared to the $H^{0}(X_1, \omega_{X_1})_{\chi_1}$, the difference lies between $m_{\chi_1}(\hat{S}_{X_1}^{\chi_1})$ and $m_{\chi_1}(\hat{S}_{X_1}^{\chi_1} \sqcup I_1^{\chi})$. The latter includes additional ramification data at intersection points. Note that when $d = 1$, we have $X = X_1$ and $I_1^{\chi} = \varnothing$. Therefore,

Hence, this formula directly generalizes the irreducible case mentioned above.

2.   Preliminaries

In this paper, we consider a finite group $G$ acting faithfully on a connected nodal curve $X$ over an algebraically closed field $k$. Let $\#G = n$ and suppose that either $\operatorname{char}(k) = p \nmid n$ or $\operatorname{char}(k) = 0$ so that $k[G]$ is semisimple. A curve means an equidimensional reduced projective scheme of finite type of dimension $1$ over $k$.

2.1.   Regular differentials on nodal curves

Let $X$ be a connected nodal curve. Since $X$ is a projective variety, the dualizing sheaf $\omega_X$ always exists and can be explicitly described. Generally speaking, if $X\subseteq \mathbb{P}_k^N$ with codimension $r$, then the dualizing sheaf of $X$ is given by $\omega_X = \mathcal{E} x t_{\mathbb{P}_k^N}^r(\mathcal{O}_X, \omega_{\mathbb{P}_k^N})$ [7, Ⅲ.7.5]. Once the dualizing sheaf exists, it is unique together with its trace map [7, Ⅲ.7.2].

It can be seen that the construction above involves a choice of ambient space $\mathbb{P}_k^N$. However, for the nodal curve, there is a simpler description based on its normalization.

Let $\alpha: \hat{X} \rightarrow X$ be the normalization of reduced curves [11, 7.5.1], namely we have $\hat{X} = \coprod_{1 \leq i \leq d} \hat{X}_i$, where each $\hat{X}_i$ is the normalization of the irreducible component $X_i$ in $X$. Each $\hat{X}_i$ is a smooth curve, so $\omega_{\hat{X}} = \Omega_{\hat{X}}$ is the canonical sheaf of $X$. Let $\hat{S}_X\subset \hat{X}$ be the preimage of the nodes in $X$, and then we define the sheaf of regular differentials on an open subset $V\subseteq X$ by

We can see that $\omega_X^{\text {reg}}$ is an $\mathcal{O}_X$-module, and we call its global sections the regular differentials on $X$. We say a meromorphic differential $\varphi_0$ in $\hat{X}$ with at worst simple poles is regular at a point $P\in X$ if $\sum_{\hat{P} \in \alpha^{-1}(P)} \operatorname{Res}_{\hat{P}}(\varphi_0) = 0$, namely the residue relation holds at $P$.

Remark 2.1. The definition and properties of residues for differentials on curves can be found in [7, Ⅲ.7.14]. Note that when $X$ is a smooth curve, the regular differentials on $X$ are exactly the holomorphic differentials. Therefore, regular differentials are the natural generalization of holomorphic differentials.

In fact, the following fact holds for dualizing sheaves on nodal curves:

Theorem 2.2. Let $X$ be a nodal curve, and then we have the canonical isomorphism $\omega_X^{reg} \simeq \omega_X$.

This is a highly nontrivial result. In the exercises of [6, 3.A], one can verify the universal property of the dualizing sheaf by endowing $\omega_X^{\text{reg}}$ with a trace map. For further verification on the compatibility, the readers are referred to the detailed proof in [3, 5.2].

In summary, the dualizing sheaf on a nodal curve has a precise characterization that depends only on the structure of meromorphic differentials with at worst simple poles in $\Omega_{\hat{X}}(\hat{S}_X)$.

It is known that $H^0(X, \omega_X)$ is a $k$-vector space of dimension $p_a(X)$, the arithmetic genus of $X$. Since the finite group $G$ acts on $X$, both the rational function field $K(X)$ and $H^0(X, \omega_X)$ are naturally (right) $k[G]$-modules. Every $1$-dimensional representation is its own character. Our goal is to compute the multiplicity of every character $\chi:G\rightarrow k^{\times}$, that is, the dimension of the eigenspace $H^0(X, \omega _X)_{\chi}: = \{\varphi \in H^0(X, \omega _X)\mid \sigma \varphi = \chi(\sigma) \varphi, \; \forall \sigma \in G\}$ over $k$.

2.2.   Hilbert's Theorem 90

For the sake of discussion, let $X$ be smooth for the rest of this section. Let $G\subseteq \operatorname{Aut}(X)$ be a finite group of automorphisms of order $n$. Then the quotient map $\pi:X\rightarrow Y: = X/G$ is a Galois covering, i.e., $K(X)/K(Y)$ is a Galois field extension, where $K(X)$ and $K(Y)$ are the rational function fields of the corresponding curves.

Proposition 2.3. Let $\chi:G\rightarrow k^{\times}$ be a character, and then there exists a rational function $f_\chi \in K(X)^{\times}$ such that $\sigma f_\chi = \chi(\sigma) f_\chi, \; \forall \sigma \in G$.

This result is a special case of the following theorem (in [13, 5.23]):

Theorem 2.4. Let $E/F$ be a Galois field extension, and let $G = Gal(E/F)$. Then $H^1\left(G, E^{\times}\right) = 0$.

Remark 2.5. (NOTES below Corollary 5.25 in ^[13]). This theorem is a generalization of the famous Hilbert's Theorem 90, which was first discovered by Kummer in the case of $\mathbb{Q}[\xi_p]/\mathbb{Q}$, and later generalized by Emmy Noether. This theorem and its various generalizations are all referred to as Hilbert's Theorem 90.

Here we only prove Proposition 2.3. Before that, we first prove a lemma:

Lemma 2.6 (Dedekind's independence theorem). Let $F$ be a field, and $G$ be a group. Then any finite number of different group homomorphisms $\chi_1, \ldots, \chi_m:G \rightarrow F^{\times}$ are linearly independent over $F$, i.e.,

Proof. We use induction on $m$. The statement is obvious when $m = 1$. Assume it holds for $m-1$. Suppose there exist $a_i \in F$ such that

Next, we prove $a_i = 0$. Without loss of generality, suppose for some $g \in G$, $\chi_1(g)\neq \chi_2(g)$, and then we have

Subtracting the first equation multiplied by $\chi_1(g)$ from this equation, we get

By induction, $a_i^{\prime} = 0, \; i = 2, \cdots, m$. Since $\chi_2(g)-\chi_1(g) \neq 0$, $a_2 = 0$, thus we have

By the induction hypothesis, the rest of the $a_i = 0$. □

Proof of Proposition 2.3. Consider the mapping

By the above lemma, this mapping is not zero, so there exists a rational function $g \in K(X)$ such that

Note that, for $\forall \sigma \in G$, we have

Thus $f_\chi: = f^{-1}$ is the desired function. □

2.3.   Ramification data on curve quotients

The quotient map $\pi:X\rightarrow Y$ is also a ramified cover. Let $e_P$ be the ramification index at $P\in X$, and then we have the ramification divisor

For a divisor $D = \sum a_i P_i \in \operatorname{Div}(X)$, define $\pi_*D \in \operatorname{Div}(Y)$ by

If $D = \sum a_i Q_i \in \operatorname{Div}(Y)$ is a divisor and $r \in \mathbb{R}$, then define $\left\lfloor rD \right\rfloor \in \operatorname{Div}(Y)$ by

where $\lfloor r a_i \rfloor$ denotes the greatest integer $\leq r a_i$. Define $\pi^*D \in \operatorname{Div}(Y)$ by

Proposition 2.7 (Kani ^[9]). Let $G$ be a finite group (of order $n$) acting on a smooth curve $X$ with $R_{\pi}$ the ramification divisor of $\pi: X \rightarrow Y = X/G$. Suppose $D \in \operatorname{Div}(X)$ is a $G$-invariant divisor, and then for the trivial character $\chi = 1_G$, we have

For any character $\chi$, let $f_\chi \in K(X)^*$ be such that $\sigma f_\chi = \chi(\sigma) f_\chi$ for all $\sigma \in G$ (whose existence is guaranteed by Hilbert's Theorem 90). Then

Proof. (More details are given here than in ^[9].) First, it is easily deduced from the definition that $\pi^* \pi_* D = nD$ since $D$ is $G$-invariant.

For (2.2), we have $D \geq \pi^*\left\lfloor n^{-1} \pi_* D\right\rfloor$ and hence $H^{0}(X, \mathcal{ O}_X(D))^G \supseteq \pi^* H^{0}(Y, \mathcal{O}_Y \left\lfloor n^{-1} \pi_* D\right\rfloor)$. Conversely, if $f \in H^{0}(X, \mathcal{ O}_X(D))^G$, then $f = \pi^* e$ with some $e \in K(Y)$. Hence

which implies $\left(e\right)+\left\lfloor n^{-1} \pi_* D\right\rfloor \geq 0$.

To prove (2.3), fix a meromorphic differential $0 \neq \varphi \in \Omega(Y)$, which exists by Riemann-Roch provided that the pole multiplicities are sufficiently high. By

we have

Finally, (2.4) and (2.5) for general $\chi$ follow from

Note that the divisor $\left(f_\chi\right)$ is $G$-invariant, so it suffices to repeat the discussion above with $D$ replaced by $D+\left(f_\chi\right)$. □

2.4.   Ramification modules

The constructions in this section are based on Kani's work^[9]. Fix a point $P \in X$, and let $G_P: = \{\sigma \in G \mid \sigma(P) = P\}$ be the stabilizer subgroup of $G$ at $P$, which is a cyclic group of order $e_P$. Then there is a unique character $\theta_P: G_P \rightarrow k^{\times}$ such that for any $f\in K(X)^{\times}$,

where $v_P$ denotes the valuation at $P$ and $\mathfrak{m}_P$ the maximal ideal of the local ring $\mathcal{ O}_P$.

Set

^*Here $d \cdot \theta_P^d$ means $\oplus^{d} \theta_P^d$.

Definition 2.8. Let $Bl(Y)$ be the branch locus of $\pi:X\rightarrow Y$, namely the subset of all branch points in $Y$. For a point $Q\in Y$, we define the ramification module of $Q$ by

and the ramification module of $\pi$ by

Note that this is a finite sum because $R_{G, Q} = 0$ for $Q\notin Bl(Y)$.

Let $\chi:G\rightarrow k^{\times}$ be a character and $f_\chi\in K(X)_{\chi}$ as in Proposition 2.3. Since $\chi^n = 1_G$, we have $f_{\chi}^n \in k(X)^G = \pi^* k(Y)$. Write $\left(f_{\chi}^n\right) = \pi^*(n A+B)$ where $A, B \in \operatorname{Div}(Y)$ and $\lfloor n^{-1} B\rfloor = 0$. Note that $\operatorname{Supp}(B)\subseteq Bl(Y)$, so we write $B = \sum_{Q\in Bl(Y)} b_Q Q$. By definition, we have

$^{\dagger}$In the expression $v_{Q}(f_{\chi}^n)$, the function $f_{\chi}^n$ is regarded as a rational function on $Y$. Consequently, we have $v_{P}(f_{\chi}^n) = e_Qv_{Q}(f_{\chi}^n)$.

where $\langle r\rangle = r-\lfloor r \rfloor$ denotes the fractional part of $r$.

The following lemma shows that this $B$ is independent of the choices of $f_{\chi}$:

Lemma 2.9. Let $\chi: G \rightarrow k^{\times}$ be a character. Then for any $Q\in Bl(Y)$, we have

Proof. Let $P \in \pi^{-1}(Q)$. Then by Frobenius reciprocity, we have

Note that $\theta_P^d$ runs through are all the irreducible representations of $G_P$, and hence we have

for $0 \leq a < e_P$. Choose a generator $\sigma$ of $G_P$, and then by the definition of $f_\chi$, we have

Furthermore, by the definition of $\theta_P$, we have

which implies $a\equiv v_P(f_{\chi})(\operatorname{mod} \; e_P)$ since $\theta_P(\sigma)$ has order $e_P$ in $k^{\times}$. Finally,

and hence we have

□

3.   Irreducible nodal curves

Here is a basic result on the nodal curve quotient [11, 10.3.48]:

Proposition 3.1. Let $X$ be a reduced nodal curve, and $G\subseteq \operatorname{Aut}(X)$ be a finite automorphism group on $X$. Then the quotient curve $Y = X / G$ is a nodal curve. More precisely, let $P\in X$ be a closed point, and $Q$ be its image in $Y$. Then we have the following results:

$(a)$ If $X$ is smooth at $P$, then $Y$ is smooth at $Q$;

$(b)$ If $P$ is an ordinary double point on $X$, then $Q$ is either a smooth point or an ordinary double point.

Let $X$ be an irreducible nodal curve in this section.

3.1.   The $G$-invariant differentials

Let $G$ be a finite group acting on an irreducible nodal curve $X$ and $Y = X/G$ the quotient curve, which is also an irreducible nodal curve. We want to study the $G$-invariant space of regular differentials $H^{0}(X, \omega_{X})^G$. It is a classical fact that:

Proposition 3.2. If $X$ is smooth, then

Proof. With the notations of Section 2.3, let $e_Q: = e_{P}$ for any $P\in \pi^{-1}(Q)$. Note that

By Proposition 2.7 (2.3), we have

which is of dimension $g_Y$, the geometric genus of $Y$. □

Here comes a natural question: For the covering $\pi:X\rightarrow Y$ of irreducible nodal curves, do we still have the equality

The answer is yes.

Consider the normalizations $\hat{X} \rightarrow X$ and $\hat{Y}\rightarrow Y$, respectively. We have a canonical isomorphism $\hat{X}/G = \hat{Y}$, so there is a commutative diagram:

This induces the corresponding morphisms of differentials

The lower row is obtained by $\hat{\pi}^{-1}(\hat{S}_Y)\subseteq \hat{S}_X$, since $X\rightarrow Y$ maps smooth points to smooth points.

Lemma 3.3. There is a canonical inclusion

that makes (3.2) commute.

Proof. We say $\{P_1, P_2\} \subseteq \hat{S}_X$ is a pair if the two points are precisely the preimages of the same node in $X$. In this context, a differential $\varphi \in H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X))$ is regular if and only if

for every pair $\{P_1, P_2\}$.

In this situation, we say that $\{P_1, P_2\}$ is the preimage of the corresponding node, denoted by $P$. Note that for any pair $\{P_1, P_2\}$, the ramification indices are equal, i.e.,

since the orbits of both points have the same cardinality, namely, $\#\{\sigma(P) \in X \mid \sigma \in G\}$.

Now, consider a regular differential $\varphi_Y \in H^0(Y, \omega_Y)$. For any pair $\{P_1, P_2\} = \alpha^{-1}(P) \subseteq \hat{S}_X$, we have

Moreover, for any point $P_0 \in \hat{X}$, the pullback satisfies

Thus, applying these equalities for the pair $\{P_1, P_2\}$, we obtain

This shows that $\hat{\pi}^*\varphi_Y$ is regular at $P$, and hence is a regular differential in $X$. □

Furthermore, we have

Lemma 3.4. For the left column of (3.2), we have

Proof. All we need is to show a $G$-invariant regular differential has no poles at $\hat{S}_X-\hat{\pi}^{-1}(\hat{S}_Y)$. Suppose $\hat{S}_X-\hat{\pi}^{-1}(\hat{S}_Y)\neq \varnothing$, otherwise there is nothing to prove.

Given a pair $\{P_1, P_2\}\subseteq \hat{S}_X-\hat{\pi}^{-1}(\hat{S}_Y)$, we know that the points of $\hat{S}_X-\hat{\pi}^{-1}(\hat{S}_Y)$ are mapped to the smooth locus of $Y$, and hence there exists a $\sigma\in G$ such that $\sigma(P_1) = P_2$. So for any regular differential $\varphi\in H^{0}(X, \omega_{X})^G$, we have $\operatorname{Res}_{P_1}\varphi = \operatorname{Res}_{P_1}\sigma \varphi = \operatorname{Res}_{\sigma(P_1)} \varphi = \operatorname{Res}_{P_2}\varphi.$ As $\operatorname{Res}_{P_1}\varphi = -\operatorname{Res}_{P_2}\varphi$ by definition, it forces that $\operatorname{Res}_{P_1}\varphi = \operatorname{Res}_{P_2}\varphi = 0$, which implies $\varphi$ has no poles at $\{P_1, P_2\}$. □

Now we can give a positive answer to (3.1).

Theorem 3.5. With the notations above, the lower row of the canonical commutative diagram

is an isomorphism. Moreover, both $H^{0}(X, \omega_{X})^G$ and $H^{0}(Y, \omega_{Y})$ are the subspaces of log differentials satisfying the residue relations, so we have the isomorphism for the upper row. In particular, we have $\mathrm{dim}_k H^{0}(X, \omega_X)^G = p_a(Y)$.

Back to the smooth case, we have the following:

Proposition 3.6. Suppose $\pi:X\rightarrow Y$ is the quotient morphism of smooth curves and $S\subset Y$ is a finite set. Then

is an isomorphism.

Proof. By Proposition 2.7 (2.3), we have

^*Here and there, we adopt the convention that every finite subset is interpreted as an effective divisor when needed.

First, note that for the divisor $n^{-1}\pi_*\pi^{-1}(S)$, the coefficient corresponding to a point $Q\in S$ is exactly $1/e_Q$. Now, consider

and analyze the coefficient $a_Q$ for each prime divisor $Q$ by considering three cases:

● If $Q\in Bl(Y)-S$ (i.e., $Q$ belongs to the branch locus but is not in $S$), then

● If $Q\in S- Bl(Y)$ (i.e., $Q$ is in $S$ but not in the branch locus), then

● If $Q\in S\cap Bl(Y)$ (i.e., $Q$ is both in $S$ and in the branch locus), then

Since these are the only possibilities, it follows that $\left\lfloor n^{-1}\pi_*\bigl(\pi^{-1}(S)+R_\pi\bigr) \right\rfloor = S.$ □

Proof of Theorem 3.5. Apply Proposition 3.6 to $\hat{\pi}:\hat{X} \rightarrow \hat{Y}$ and $\hat{S}_Y\subset \hat{Y}$. □

3.2.   Chevalley-Weil formula for irreducible nodal curves

Let $\chi$ be a character of $G$, and $\alpha:\hat{X} \rightarrow X$ the normalization. With the notations in (3.2), consider the embedding

We want to determine the image of $H^{0}(X, \omega_{X})_{\chi}$ in $H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X))_{\chi}$.

Proposition 3.7. Suppose that $Y$ is smooth. Define

Then the image of $H^{0}(X, \omega_{X})_{\chi}$ in $H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X))_{\chi}$ is equal to $H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X^{\chi}))_{\chi}$. So we have an isomorphism

We call $\hat{S}_X^{\chi}$ the $\textrm{singular $ \chi $-set}$ of $\pi$.

Proof. Let $\varphi_{0}\in H^{0}(X, \omega_{X})_{\chi}$ a regular differential. If it has poles on a pair $\{P_1, P_2\}\subseteq \hat{S}_X$, then there is some $T\in G_P$ such that $T(P_1) = P_2 {}^{\dagger}$, which is guaranteed by the smoothness of $Y$. So we have

$^{\dagger}$Hence $T(P_2) = P_1$ by symmetry.

which implies $\chi(T) = -1$ and $P_1, P_2\in \hat{S}_X^{\chi}$. In particular, when $\hat{S}_X^{\chi} = \varnothing$, we see that any regular differential in $H^{0}(X, \omega_{X})_{\chi}$ has no pole on $\hat{X}$.

Conversely, assume $\varphi \in H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X^{\chi}))_{\chi}$ and $\hat{S}_X^{\chi}\neq\varnothing$. Let $\{P_1, P_2\}\subseteq \hat{S}_X^{\chi}$ be a pair. By definition, there is some automorphism $T\in G_P$ such that $T(P_1) = P_2$ and $\chi(T) = -1$. Hence

This means $\varphi$ is regular. □

Remark 3.8. If $\chi = 1_G$ is the trivial representation, then $\chi(\sigma)\neq -1$ for any $\sigma\in G$, and hence $\hat{S}_X^{1_G} = \varnothing$. By Propositions 3.7 and 3.2, we obtain

This conclusion is consistent with Proposition 3.5 in the case that $Y$ is smooth, in which case $\hat{S}_Y = \varnothing$.

Remark 3.9. Here we explain why the condition of $Y$ being smooth is needed. Suppose the meromorphic differential $\varphi_0 \in H^{0}(\hat{X}, \Omega_{\hat{X}}(S))$ has a simple pole at some point $P_1 \in S$, i.e., $\operatorname{Res}_{P_1}\varphi_0 \neq 0$. If $\varphi_0 \in H^{0}(X, \omega_{X})_{\chi}$, then $\varphi_0$ is regular at $\alpha(P_1)$ ($\alpha$ is the normalization), so that there exists a pair $\{P_1, P_2\}$ on $\hat{X}$ satisfying the following two conditions:

(a) $v_{P_2}(\varphi_0) = v_{P_1}(\varphi_0) = -1$;

(b) $\operatorname{Res}_{P_2}\varphi_0 = -\operatorname{Res}_{P_1}\varphi_0$.

For condition (a), generally, we cannot determine the value $v_{P_2}(\varphi_0)$ from $v_{P_1}(\varphi_0)$. However, when $Y$ is smooth, we have the following commutative diagram:

Here $\hat{\pi}$ is a ramified cover of smooth curves. Since $P_1, P_2$ are mapped to the same point in $Y$ (through $X$), there exists an automorphism $\tau$ that permutes the pair $\{P_1, P_2\}$ and satisfies $\tau\varphi_0 = \chi(\tau)\varphi_0$. Consequently, we must have

and

For condition (b), under the hypothesis $Y$ being smooth, it is equivalent to say $\chi(\tau) = -1$ for some (and hence any) automorphism $\tau$ permuting $\{P_1, P_2\}$. Furthermore, this criterion works for the intersection points in a nodal curve with several irreducible components (see Section 4.2).

Assume $Y = X/G$ is smooth for the rest of this section, and then $\hat{\pi}:\hat{X} \rightarrow Y$ is the induced ramified cover of $\pi:X \rightarrow Y$. Now we compute the dimension of $H^{0}(\hat{X}, \Omega_{\hat{X}}(\hat{S}_X^{\chi}))_{\chi}$ through Proposition 2.7.

Let $f_\chi$ be a rational function on $\hat{X}$ such that $\sigma f_\chi = \chi(\sigma) f_\chi, \; \forall \sigma \in G$. Set

By Proposition 2.7 (2.5), we have

At this point, we reduce the problem to calculating the dimension of $H^{0}(Y, \Omega_Y(D_{\chi}))$.

Using the Riemann-Roch theorem on $Y$, we have

For $\mathrm{dim}_k H^{0}(Y, \mathcal{O}_Y(-D_{\chi}))$, we have:

Lemma 3.10. The space $H^{0}(Y, \mathcal{O}_Y(-D_{\chi}))$ vanishes except when $\chi = 1_G$, and in this case, we have $\mathrm{dim}_k H^{0}(Y, -D_{1_G}) = 1$.

Proof. We will prove the following:

$1)$ If $\hat{S}_X^{\chi}\neq \varnothing$, then $\operatorname{deg} D_{\chi} > 0$.

$2)$ The divisor $D_{\chi}$ is principal if and only if $\chi = 1_G$; in this case, $\hat{S}_X^{\chi} = \varnothing$.

Recall that we have

where $a_Q = v_P(f_\chi)$, $\forall P\in \hat{\pi}^{-1}(Q)$. Note that

Therefore, we have the inequalities

Now, write

If $\hat{S}_X^{\chi}\neq \varnothing$, then there exists some point $Q'$ such that $c_{Q'}\geq 1$. Hence, we can write

Thus we have $\operatorname{deg} D_{\chi} > 0$ provided $\hat{S}_X^{\chi}\neq \varnothing$, and consequently

Next, assume that $\chi = 1_G$. It follows that $\hat{S}_X^{\chi} = \varnothing$ and $f_\chi = \hat{\pi}^* h$ for some rational function $h\in K(Y)$. Then we have

which shows that $D_{\chi}$ is principal. Therefore, in this case, $\mathrm{dim}_k H^{0}(Y, \mathcal{O}_Y(-D_{\chi})) = 1$.

Conversely, suppose that $D_{\chi}$ is a principal divisor; that is, there exists some rational function $h\in K(Y)$ such that $D_{\chi} = \left(h\right)$. Then $\operatorname{deg} D_{\chi} = 0$, which forces $\hat{S}_X^{\chi} = \varnothing$. Then, by the previous calculations,

This chain of equalities implies that

which means that $e_Q\mid a_Q$. Therefore,

Applying $\hat{\pi}^*$ to both sides, we obtain

which implies that $\left(f_\chi\right) = (\hat{\pi}^* h)$. In other words, $f_\chi \in K(X)^G$, i.e., $\chi = 1_G$. □

Now we want to calculate $\operatorname{deg} D_{\chi}$.

Definition 3.11. Let $S\subseteq \hat{X}$ be a finite subset stable by $G$, and define

where $R_{G, Q}$ and $R_G$ are ramification modules of $\hat{\pi}:\hat{X} \rightarrow Y$.

Lemma 3.12. We have $\operatorname{deg} D_{\chi} = m_{\chi}(\hat{S}_X^{\chi})$, which is independent of the choice of $f_{\chi}$.

Proof. Let $\left(f_{\chi}^n\right) = \hat{\pi}^*(n A+B)$ as in the discussion before Lemma 2.9 satisfying $\lfloor n^{-1} B\rfloor = 0$. Then we have

Note that

So we decompose $n^{-1}\hat{\pi}_*(\hat{S}_X^{\chi}) = U+V$ into two parts according to whether the point is branched, so that $V$ is a $\mathbb{Q}$-divisor which is supported on the branch locus of $\hat{\pi}$, i.e., $\operatorname{Supp}(V) = Bl(Y)\cap \hat{\pi}(\hat{S}_X^{\chi})$. So $U$ is an integer divisor with all coefficients equal to $1$. Now we have

Let $B = \sum_{Q} b_Q Q$ (supported on $Bl(Y)$) and according to $\operatorname{deg}B = -\operatorname{deg}nA$, we get

By (2.6) in Lemma 2.9, we have $b_Q = \langle \chi, R_{G, Q}\rangle_G$, so

□

We summarize the discussion in this section.

Let $X$ be an irreducible nodal curve over $k$, $G$ be a finite automorphism group of order $n$ on $X$, and $Y = X/G$ be a smooth curve. Let $\hat{\pi}:\hat{X} \rightarrow Y$ be the ramified cover induced by the normalization $\pi: X \rightarrow Y$, and $\hat{S}_X\subseteq \hat{X}$ be the preimage of the singular points of $X$ on $\hat{X}$. Denote $R_G$ the ramification module of $\hat{\pi}$, and $R_{G, Q}$ the ramification module of $Q\in Y$.

Theorem 3.13 (The Chevalley-Weil formula on irreducible nodal curves). Let $\chi:G\rightarrow k^{\times}$ be a character. If the quotient curve $Y = X/G$ is smooth, then

Here $\hat{S}_X^{\chi}$ is the singular $\chi$-set of $\pi$ as in (3.5), and in Definition 3.11,

In particular, when $\chi = 1_G$, we have $\hat{S}_X^{\chi} = \varnothing$ and $\operatorname{dim}_k H^{0}(X, \omega_{X})^G = g_Y.$ This is the special case that $\hat{S}_Y = \varnothing$ in Proposition 3.5.

Example 3.14 (Hyperelliptic stable curves). We call a stable curve $C$ a hyperelliptic stable curve if there exists a $2$-order automorphism $J:C\rightarrow C$ such that $C/ \langle J \rangle = \mathbb{P}^1$. Such an automorphism of $C$ is called an involution.

Let $C$ be an irreducible hyperelliptic stable curve with $N$ ($\geq 1$) nodes, and $\hat{C}$ be the normalization of $C$ with genus $g$. Then $\hat{\pi}:\hat{C}\rightarrow \mathbb{P}^1$ has $2g+2$ fixed points, which are also all the ramification points of $\hat{\pi}$, with ramification index all equal to $2$. So the number of branch points on $\mathbb{P}^1$ is $\#Bl(\mathbb{P}^1) = 2g+2$ and no branch point lies in $\hat{\pi}(\hat{S}_C)$.

Note that the Galois group $G = \langle J \rangle\cong \mathbb{Z}_2$ has two characters $1_G$ and $\chi^-$, where $\chi^-(J) = -1$.

For $\chi = 1_G$, we have $\operatorname{dim}_k H^{0}(C, \omega_{C})^G = g({\mathbb{P}^1}) = 0.$

For $\chi^-$, we have $\hat{S}_C^{\chi^-} = \hat{S}_C$, so $s_{\chi^-} = N$. The induced character at any branch point $P$ is

Therefore $R_{G, Q} = \theta_P$, so for all $Q\in Bl(\mathbb{P}^1)$, we have $\langle \chi^-, R_{G, Q}\rangle_G = 1$, and then $\langle \chi^-, R_G\rangle_G = 2g+2.$ We calculate

Finally, we get $\operatorname{dim}_k H^{0}(C, \omega_{C})_{\chi^-} = g({\mathbb{P}^1})-1+ m_{\chi^-}(\hat{S}_C^{\chi^-}) = p_a(C).$

4.   Nodal curves with several irreducible components

In this section, let $X$ be a connected nodal curve with several irreducible components. Suppose that $X$ admits a finite automorphism group $G$, and let $\pi: X \rightarrow Y = X/G$ be the quotient morphism. Consequently, $Y = X/G$ is also a connected nodal curve.

We will prove that $H^{0}(X, \omega_X)^G = \pi^* H^{0}(Y, \omega_Y)$ and then generalize Theorem 3.13 to the Chevalley-Weil formula for connected nodal curves, assuming the smoothness of $Y$ for the same reasons stated in Remark 3.9.

Due to the abundance of symbols, in the following, we will replace some round brackets with square brackets, such as using $H^{0}(Y, \omega_{Y}[I_{Y}])$ instead of $H^{0}(Y, \omega_{Y}(I_{Y}))$.

4.1.   Reduction

Let $Y = \cup_{j} Y_j$ be the decomposition into irreducible components. Then

is the normalization at the intersection points. Let $I_{Y_j}$ be the set of intersection points of $Y_j$ with other components, and we call it the intersection locus of $Y_j$. Note that these points are nodes in the whole $Y$, but smooth points in each $Y_j$.

Then we can decompose the regular differential of $Y$ by the irreducible components:

Furthermore let $\hat{Y}_j \rightarrow Y_j$ be the normalization. Then we have

Now let $X'_j: = \pi^{-1}(Y_j)$ be the preimage of an irreducible component. This is a nodal curve but could be disconnected and we have^*

^*If a curve $X$ decomposes as $X = X_1 \sqcup X_2$, then the space of differentials decomposes as $H^0(X, \omega_X) = H^0(X_1, \omega_{X_1})\oplus H^0(X_2, \omega_{X_2}).$

where $I_{X'_j} = \pi^{-1}(I_{Y_j})$. In fact, the information of the eigenspace $H^0(X'_j, \omega_{X'_j})_{\chi}$ of the regular differentials is contained in any single irreducible component.

Let $X$ be a (possibly disconnected) nodal curve decomposed into its irreducible components

and suppose that $Y = X/G$ is irreducible. Then the action of $G$ on the set of irreducible components $\{X_1, \dots, X_d\}$ is transitive, and in particular, all these irreducible components are isomorphic. Let $S \subset X$ be a $G$-invariant divisor, and set $S_i : = S|_{X_i}$. Consequently, $G$ acts on

More precisely, for any $\sigma \in G$ we define the permutation of the indices by setting $\sigma(i)$ via the relation $\sigma(X_i) = X_{\sigma(i)}$. Then for any $\varphi \in H^{0}(X_i, \omega_{X_i}[S_i])$ we have:

$^{\dagger}$ $\sigma$ acts by pullback on the differentials (that is, if $\sigma:X_j\rightarrow X_i$ and $\varphi\in H^{0}(X_i, \omega_{X_j})$, then $\sigma \varphi: = \sigma^* \varphi \in H^{0}(X_j, \omega_{X_j})$.

Let $G_i : = \{\sigma \in G \mid \sigma(X_i) = X_i\}$ denote the stabilizer subgroup of $X_i$. Then we have:

Proposition 4.1. Given a character $\chi:G\rightarrow k^{\times}$ with restriction $\chi_i: = \chi|_{G_i}:G_i\rightarrow k^{\times}$ to $G_i$, we have a canonical projection

which is an isomorphism.

Proof. We show this holds for $p_1$.

Suppose

Then for any $\sigma \in G$ we have

Since $G$ acts transitively on the set $\{X_1, \dots, X_d\}$, for each $i$ there exists some $\sigma_i: X_i \to X_1$. Hence, we can express the tuple as

which shows that the tuple $(\varphi_i)_i$ is completely determined by its first component $\varphi_1$. This proves that the projection

is injective. It remains to show that $\varphi_1$ lies in the $\chi_1$-eigenspace. For any element $\tau \in G_1$ (the stabilizer of $X_1$), we have $\tau(1) = 1$ and $\chi(\tau) = \chi_1(\tau)$. Therefore,

which confirms that $\varphi_1$ indeed belongs to $H^{0}(X_1, \omega_{X_1}[S_1])_{\chi_1}$.

Conversely, let

and for each $i$ choose an isomorphism $\sigma_i : X_i \to X_1$ as before. We need to show that

defines an element of

First, note that if we choose two different isomorphisms $\sigma_i, \tau_i : X_i \to X_1$, then $\tau_i^{-1} \sigma_i$ is an element of the stabilizer $G_1$ of $X_1$. Since $\varphi_1$ lies in the $\chi_1$-eigenspace, we have

Because the character $\chi$ restricts on $G_1$ to $\chi_1$, this can be rewritten as

Multiplying both sides on the left by $\chi(\sigma_i)^{-1}\chi(\tau_i)\tau_i$ yields

Thus, the section

is independent of the particular choice of the isomorphisms $\sigma_i$.

Next, let $\tau \in G$ be arbitrary. We may assume that $\tau$ sends the index $j$ to $i$, that is, $\tau(j) = i$. Then, by the definition of the action, we have

Since $\tau \sigma_i$ and $\sigma_j$ are both isomorphisms from $X_j$ to $X_1$, we have

Thus,

It follows that

In conclusion, we obtain that $p_1$ is an isomorphism. □

Let $X_j$ be a chosen irreducible component of $X'_j$. By Proposition 4.1, there are immersions

and

where $\chi_j = \chi|_{G_j}$. Consequently, we obtain the chain of inclusions

Here:

● $G_j = \{\sigma\in G\mid \sigma(X_j) = X_j\}$ and $|G_j| = |G|/d_j$ if $X'_j$ has $d_j$ irreducible components.

● $I'_{j} = I_{X'_j}\cap X_j$.

● $I_{X_j}$ is the set of intersection points of $X_j$ with other components in $X'_j$.

● $\hat{S}_{X_j}$ is the preimage of nodes in the normalization $\hat{\alpha}_j:\hat{X}_j\rightarrow X_j$.

Now we consider the $G$-invariant regular differentials. Suppose $\chi = 1_G$ and consider the following diagram as (3.2):

Lemma 4.2. There is a canonical inclusion

that makes the diagram above commute.

Proof. Let

be a regular differential on $Y$ and consider its image in the lower left-hand corner:

We need to verify that the residues of $\hat{\pi}^*\varphi_Y$ satisfy the relation

where $\{P_1, P_2\}$ is a pair in $\hat{X}$, i.e., $P_1$ and $P_2$ are points lying over the same node $P$ in $X$.

Without loss of generality, assume that $P_1\in \hat{X}_1$, the normalization of $X_1$, which is a chosen irreducible component of $X'_1$. Then

where $e_{P_1}$ is the ramification index of $P_1$ in the cover of smooth curves

According to Lemma 3.3, the remaining task is to establish the equality

To see this, let $l_P$ denote the total number of elements in the orbit of the point $P$ (which is common to both $P_1$ and $P_2$) under the action of the group $G$. Then the orbit of $P_1$ has exactly $l_P/d_1$ elements in $\hat{X}_1$, where $d_1$ is the number of irreducible components in $X'_1$. Hence, by the very definition of the ramification index we obtain

By the same argument to $P_2$, we also have

This completes the proof. □

The same argument as Lemma 3.4 tells us

With the notations above, we have:

Theorem 4.3. The upper and lower row of the canonical commutative diagram

are both isomorphisms. In particular, we have $\mathrm{dim}_k H^{0}(X, \omega_X)^G = p_a(Y)$.

Proof. By Proposition 3.6, we have

So we have the isomorphism for the lower row. Moreover, both $H^{0}(X, \omega_{X})^G$ and $H^{0}(Y, \omega_{Y})$ are the subspaces of log differentials satisfying the residue relations, so we have the isomorphism for the upper row. □

4.2.   Chevalley-Weil formula for connected nodal curves

Let $Y$ be smooth for the remaining part of this section. We are going to calculate $\operatorname{dim}_k H^{0}(X, \omega_{X})_{\chi}$ for general $\chi$.

Let $X = \cup^{d}_{i = 1} X_i$ be the decomposition of irreducible components, and then we can decompose the regular differential space on $X$ into each irreducible component:

where $I_i$ is the intersection locus of $X_i$.

Since $Y$ is smooth, then by the criterion in Remark 3.9, set

as those intersection points that could be the poles of $\varphi|_{X_i}$ for $\varphi\in H^0(X, \omega_X)_{\chi}$, and then we have the isomorphism

Applying Proposition 4.1 shows that

So far, our research object has been reduced to the action of the stabilizer subgroup $G_1$ on the irreducible nodal curve $X_1$.

Remark 4.4. With the notations above, note that $I_i^{\chi} = \varnothing$ when $\chi = 1_G$. For the quotient morphism $\pi_1:X_1 \rightarrow X_1/G_1 \overset{\sim}{ = }Y$, we have

This first isomorphism is by (4.9) and the second is by Theorem 3.5. This is a special case of Theorem 4.3, and we have

Now it remains to calculate $\operatorname{dim}_k H^{0}(X_1, \omega_{X_1}[I_1^{\chi}])_{\chi_1}$.

Let $\hat{\pi}_1:\hat{X_1} \rightarrow Y$ be the normalization of $\pi_1$, $Bl(Y)$ the branch locus, $R_{G_1}$ the ramification module of $\hat{\pi}_1$, and $R_{{G_1}, Q}$ the ramification module of $Q\in Y$. If we let $\# G = n$, then $n_1: = \# G_1 = n/d$.

Let $\hat{S}_{X_1}^{\chi_1}$ be the singular $\chi_1$-set of $X_1$ as (3.5), and then we have the isomorphism

by the same argument as in Proposition 3.7.

By Proposition 2.7 (2.5) again, we have

where $f_{\chi_1} \in K(X_1)^{\times}$ satisfies $\sigma f_{\chi_1} = \chi_1(\sigma) f_{\chi_1}, \; \forall \sigma \in G_1$.

Set $D_{\chi_1}: = \left\lfloor\ n_1^{-1} \pi_*\left(\hat{S}_{X_1}^{\chi_1}\cup I_1^{\chi}+(f_{\chi_1})+R_{\hat{\pi}_1}\right)\right\rfloor$. By the Riemann-Roch theorem, we have

Applying the calculation of Lemma 3.12, we have

and see that

in Definition 3.11.

Finally, through the discussion of Lemma 3.10, we get

where $\delta_{\chi} = 0$ or $1$. $\delta_{\chi} = 1$ if and only if $D_{\chi_{1}}$ is principal; in this case, $I_1^{\chi} = \varnothing$ and $\chi_1 = 1_{G_1}$. ^‡

^‡In this context, we require that $\hat{S}_{X_1}^{\chi_1}\cup I_1^{\chi}$ is empty. In particular, when $\chi_1 = 1_{G_1}$, note that $\hat{S}_{X_1}^{\chi_1} = \varnothing$; however, $I_1^\chi$ is determined by $\chi$ rather than $\chi_1$. See Example 4.6 where even though $\chi_1^- = \mathrm{id}$, $I_1^{\chi^-}$ is nonempty.

In summary, we have obtained the following:

Let $X$ be a connected nodal curve with $d$ irreducible components, and $G$ an automorphism group on $X$ of order $n$. Assume that the quotient curve $Y = X/G$ is smooth. Then there exists a canonical ramified cover $\hat{\pi}_1:\hat{X}_1 \rightarrow X_1/G_1\simeq Y$ for an irreducible component $X_1$ with $G_1 = \{\sigma\in G\mid \sigma(X_1) = X_1\}$.

Theorem 4.5 (The Chevalley-Weil formula on connected nodal curves). Let $\chi:G\rightarrow k^{\times}$ be a character. If the quotient curve $Y = X/G$ is smooth, then

where $\chi_1$ is the restriction of $\chi$ on $G_1$, and see $m_{\chi_1}(\hat{S}_{X_1}^{\chi_1}\cup I_1^{\chi})$ in (4.14) and $\delta_{\chi}$ in (4.15).

In particular, when $\chi = 1_G$, we have $\operatorname{dim}_k H^{0}(X, \omega_{X})^G = g_Y.$

Example 4.6. Let the curves $C_1\simeq C_2\simeq \mathbb{P}^1$ intersect transversely at $m > 2$ points, and consider the hyperelliptic stable curve $C = C_1\cup C_2$, with the involution $J$ which permutes $C_1$ and $C_2$. Then we have the quotient mapping $\pi: C\rightarrow C/ \langle J \rangle = \mathbb{P}^1$. The arithmetic genus of $C$ is $p_a(C) = m-1$ (see [11, Proposition 7.5.4 and Lemma 10.3.18]) and the Galois group $\langle J \rangle$ has two characters $1_G$ and $\chi^-$.

For $\chi^-$, note that the cover $\pi_1:C_1\rightarrow \mathbb{P}^1$ induced by $\pi$ is an isomorphism, so we have $\chi^-_1 = id$. Therefore $\hat{S}_{C_1}^{id} = \varnothing$ and $m_{\chi^-_1}(\hat{S}_{C_1}^{id}\cup I_1^{\chi^-}) = \#\pi(I_1^{\chi^-}) = m$. By Theorem 4.5, we have

For $\chi = 1_G$, by (4.11), we have $\operatorname{dim}_k H^{0}(C, \omega_{C})^G = p_{a}({\mathbb{P}^1}) = 0$.

5.   Conclusions

In this work, we have systematically investigated the eigenspaces of regular differentials on a nodal curve $X$ under the action of a finite automorphism group $G$. Our primary achievement is the demonstration that the dimension of the space of $G$-invariant regular differentials is the arithmetic genus of the quotient curve $X/G$, providing a non-trivial generalization of the smooth case. Furthermore, under the condition that $X/G$ is smooth, we have successfully extended the Chevalley-Weil formula to the nodal setting, obtaining an exact expression for the dimension of the eigenspace for any one-dimensional character $\chi$. This was accomplished by first addressing irreducible nodal curves and incorporating a singular $\chi$-set, then reducing the analysis on a conncected nodal curve to that on one of its irreducible components. This reduction involves adding correction terms that account for certain intersection points on that component. These explicit formulas illuminate how nodal singularities (including intersection points) contribute to the dimensions of these eigenspaces, advancing our understanding of group actions on differentials of nodal curves.

Use of Generative-AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This article is part of the author's doctoral thesis presented to Xiamen University. The author would like to thank his supervisor Wenfei Liu for his advice and support, and he is grateful to Professor Qing Liu for helpful discussions during his visit to the University of Bordeaux. This work has been supported by the NSFC (No.11971399) and by the Presidential Research Fund of Xiamen University (No.20720210006).

Conflict of interest

The author declares no competing interests.