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A much more general result is given by Mumford, in Projective varieties I, Theorem 5.22: the volume of any $r$-dimensional smooth projective variety in $\Bbb{P}^N$ (the area in your case) is its deg …

Yes. Start from a genus 2 curve $C_2$, and choose a point of order 3 in $JC_2$, giving rise to an étale $\mathbb{Z}/3$-covering $C_4\rightarrow C_2$. Then $C_4$ cannot be hyperelliptic: a $g^1_2$ on …

This never happens. Pick a point $p\in X$; the exact sequence $0\rightarrow L^{2}\rightarrow L^{2}(p)\rightarrow \mathbb{C}_p\rightarrow 0$ gives rise to an exact sequence $0\rightarrow \mathbb{C}\x …

No, this is already false if $\pi $ is a Galois covering (i.e. $Y\cong X/G$): the index is the order of the abelianized group $G_{ab}$. Indeed from the exact sequence $$\pi _1(X)\rightarrow \pi _1(Y)\ …

Let me post my comment as an answer. Take a Weierstrass point on $X$, that is, a point $P$ for which there exists a meromorphic function $f$ with a pole of order $k\leq g$ at $P$ (there always exist …

$\mathcal{M}$ is what is called a coarse moduli space. In concrete terms, this means the following: 1) As a set, $\mathcal{M}$ can be viewed as the set of isomorphism classes of stable bundles (of r …

Yes (the answer was given, then deleted, by Francesco Polizzi). If $D$ and $D'$ are the divisors of zeroes (resp. poles) of a rational function, the linear system $|D|$ has dimension $r\geq 1$ and is …

I think this is what Picard says, translated in modern algebraic geometry language: You have a curve $X$ with a map $\pi :X\rightarrow \mathbb{P}^1$. Assume for simplicity that for each $z\in \mathbb{ …

Easy in your example : the covering curve must be $\Bbb{P}^1$ (by Hurwitz formula), with an action of $(\Bbb{Z}/2)^2$. Up to conjugacy there is no choice for such an action, it must be given by the in …

This is true. Given two points $a,b\in X$ with $u(a)$ and $u(b)\neq 0$, one can choose a path from $a$ to $b$ and a determination of $\log u$ along that path, so that $\exp\int^b_a d\log u=u(b)/u(a)$. …

Answering your PS: as you point out, the complex structure $J_t$ is given in each coordinate chart by a matrix which depends real-analytically on $t$. Now you can find a neighborhood $U$ of $[0,1]$ i …

They are not isogeneous in general. For an explicit example, take the case $n=0$ and $C$ hyperelliptic, so that we have a double covering $C\rightarrow \mathbb{P}^1$ branched along a subset $B$ of $\m …

For your last question: the map from the hyperelliptic modular group to $\operatorname{Sp}(2g,\mathbb{Z}) $ is not surjective as soon as $g\geq 3$. This was proved by V. Arnold, A remark on the branc …

If I understand correctly your question (?), you want a smooth curve $C$ of genus 5 and 3 holomorphic forms $\omega _i$ with the divisors you have written down, satisfying $\ (\omega_1^2 - \omega_2^2 …

Let $V_2$ be the image of $f:V\rightarrow W$; it is a subsheaf of $W$, hence locally free. Let $W_2$ be the quotient of $W/V_2$ by its torsion subsheaf, and let $W_1$ be the kernel of the projection …