A correction to the paper "On Curves with Split Jacobian"

TL;DR

This paper corrects and clarifies the genus formula for a family of algebraic curves, providing an algebraic proof and identifying conditions under which the original formula is valid.

Contribution

It offers a corrected genus formula with an algebraic proof and specifies the conditions for its valid application, improving upon prior incomplete results.

Findings

Corrected genus formula for certain algebraic curves

Algebraic proof of the genus formula

Conditions under which the original formula applies

Abstract

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the formula given in [5] works only under certain conditions.

Equations

$$\begin{array}{rl}zw& =c_0{x}^2+c_{1}xw+c_2{w}^2,\\ y_1^r& =h_1(z,w):=z^r+c_{1,1}{z}^{r-1}w+\cdots +c_{r-1,1}z{w}^{r-1}+w^r,\\ ⋮& ⋮\\ y_s^r& =h_1(z,w):=z^r+c_{1,s}{z}^{r-1}w+\cdots +c_{r-1,s}z{w}^{r-1}+w^r,\end{array}$$

$$g({\mathcal{X}}_{r,s})=1+r^s(s(r-1)-1).$$

$$\mathrm{deg}({\omega }_{{\mathcal{X}}_{r,s}})=(s(r-1)-1)\cdot \mathrm{deg}({\mathcal{X}}_{r,s}).$$

Full text

1 Introduction and main result

Let $r\geq 2$ and $s\geq 1$ be arbitrary integers. Let $k$ be a field and denote by ${\bar{k}}$ an algebraically closed field containing $k$. Fix a system of coordinates $x,y_{1},\cdots,y_{s},w,z$ on the projective space ${\mathbb{P}}_{\bar{k}}^{s+2}$. Let ${\mathcal{X}}_{r,s}$ denotes the algebraic curve defined over $k$ in ${\mathbb{P}}_{\bar{k}}^{s+2}$ by the following equations,

[TABLE]

where $c_{\ell}\in k$, $\ell=0,1,2,$ and $c_{i,j}\in k$ for $i=1,\cdots,r$, and $j=1,\cdots,s$.

Without giving a comprehensive proof, in Proposition 4.1 (i) of [5], Yamauchi stated that the genus $g({\mathcal{X}}_{r,s})$ of the curve ${\mathcal{X}}_{r,s}$ is equal to $(r-1)(rs\cdot 2^{s-1}-2^{s}+1)$ if it is smooth and $c_{0}\not=0$. His genus formula is used extensively in Section 4 of [1] to provide an necessary and sufficient condition in terms of $r$ and $s$ such that the Jacobian of $g({\mathcal{X}}_{r,s})$ to decompose as Jacobian of superelliptic curves.

We note that the Jacobian matrix of ${\mathcal{X}}_{r,s}$ can not have a full rank if the characteristic of $k$ divides $2r$. Thus, in this note, we assume that $k$ is a field of characteristic not dividing $2r$. we shall to provide a correct formula for the genus of ${\mathcal{X}}_{r,s}$ as stated in the following theorem.

Theorem 1.1**.**

Assume that the curve ${\mathcal{X}}_{r,s}$ is smooth and $c_{0}\not=0$. Then, its genus is equal to

[TABLE]

Proof.

Let $\omega_{{\mathcal{X}}_{r,s}}$ be the canonical sheaf of ${\mathcal{X}}_{r,s}$. By the exercise (II.8.4.e) in [3], it is isomorphic to ${\mathcal{O}}(rs+2-(s+2)-1)={\mathcal{O}}(s(r-1)-1).$ Hence, we have

[TABLE]

Using the classical version of Bezout’s theorem, see Proposition 8.4 in [2] or Example 1 in page 198 of [4], the degree of ${\mathcal{X}}_{r,s}$ over $\bar{k}$ is equal to $\deg({\mathcal{X}}_{r,s})=2\cdot r^{s}$. Thus, we get that $\deg(\omega_{{\mathcal{X}}_{r,s}})=2\cdot r^{s}\cdot(s(r-1)-1).$ As a consequence of the Riemann-Roch theorem, it is well known that the degree of canonical sheaf of any algebraic curve of genus $g$ is equal to $2g-2$. For instance, see example 1.3.3 in chapter IV of [3]. Therefore, we have $2g({\mathcal{X}}_{r,s})-2=2\cdot r^{s}\cdot(s(r-1)-1)$ that leads to the desired formula. ∎

We remark that the genus formula given in [5] and here coincide in two cases, say when $r=2$ and any $s\geq 1$ as well as $r\geq 2$ and $s=1$. Hence, Theorem 4.2 in [5] and Theorem 4.3 in [1] and its consequences are true only in the above mentioned two cases and are wrong in other cases.