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Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K \, ,$$

$P'$ is not a Weierstrass point, and neither $P'$ nor its conjugate (note that $X$ is hyperelliptic) is equal to $P$ or $i(P)$?

(Thanks to abx for his answer to my previous question).

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K \, $$

and $P$ is not conjugate to $i(P)$ (note that $X$ is hyperelliptic)?

(Thanks to abx for his answer to my previous question).

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K$$

and neither $P'$ nor its conjugate (note that $X$ is hyperelliptic) is equal to $P$ or $i(P)$?

(Thanks to abx for his answer to my previous question).

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K \, ,$$

$P'$ is not a Weierstrass point, and neither $P'$ nor its conjugate (note that $X$ is hyperelliptic) is equal to $P$ or $i(P)$?

(Thanks to abx for his answer to my previous question).

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K$$

and neither $P'$ nor its conjugate (note that $X$ is hyperelliptic) is equal to $P$ or $Q$?

(Thanks to abx for his answer to my previous question).

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K$$

and neither $P'$ nor its conjugate (note that $X$ is hyperelliptic) is equal to $P$ or $i(P)$?

(Thanks to abx for his answer to my previous question).