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Let $A$ be an abelian variety. Show that every rational map $ \mathbb{A}^{1} --> A $ or $ \mathbb{P}^{1} --> A $ is constant.

I do not have an answer to this question! Maybe the lemma on how to glue sheaves [See for example Hartshorne / Algebraic Geometry / Chapter II / Exercise 1.22] helps you obtain the answer you want. You could also try to see if this gluing construction works in "simple" examples of non-affi...

Hello! Allow me to say the following: Note that $ (f)_{0} = \left\{ \mathfrak{p} \in \text{Spec}(R) \ \big| \ f \in \mathfrak{p} \right\} = \left\{ \mathfrak{p} \in \text{Spec}(R) \ \big| \ (f) = fR \subset \mathfrak{p} \right\} = V(fR) $ $ SU = \left\{ f \in R \ \big| \ (f)_{0} \cap U = \emptyset \...

Let $\mathbb{K}$ be a field and let $A$ be a finitely generated $ \mathbb{K} $-algebra. Show that the nilradical $ \sqrt{0} $ of $A$ is the intersection of the maximal ideals of $A$ (which is the Jacobson radical $ \mathfrak{R}(A) $ of $A$).

Let $\mathbb{K}$ be a field. If $A$ is a finitely generated $\mathbb{K}$-algebra and if $\mathfrak{m}$ is a maximal ideal of $A$, then show that $A/\mathfrak{m}$ is a finite algebraic extension of $\mathbb{K}$.

Let $X$ be a complete variety and let $Y$ be an affine variety. Show that any morphism $ \varphi \ \colon X \longrightarrow Y $ is constant.

Examine whether the following Riemann Surfaces are biholomorphic : $ \mathbb{H} \, / < z \mapsto z + 2 > $ and $ \mathbb{H} \, / < z \mapsto z + 3 > $. $ \mathbb{H} \, / < z \mapsto 2z > $ and $ \mathbb{H} \, / < z \mapsto 3z > $. where by $ < z \mapsto \frac{az+b}{cz+d} > $ we mean the subgroup of ...

Definition : An annulus is a Riemann surface $A$ with fundamental group $ \pi_{1}(A) \cong \mathbb{Z} $. Show that an annulus $A$ is biholomorphic to one of the following Riemann surfaces: the punctured disc $ \mathbb{D}^{*} $ the punctured plane $ \mathbb{C}^{*} $ a round annulus $ A_{R} = \left\{...

Let $X$ be a closed Riemann surface of genus $g \geq 2$. Show that the universal covering surface $\tilde{X}$ of $X$ is biholomorphic to the upper half-plane $\mathbb{H}$.

Denote by $A_{R}$ the round annulus \[ \left\{ \, z \in \mathbb{C} \ \big| \ 1<|z|<R \, \right\} \]If $r \neq s$, are the annuli $A_{r}$ and $A_{s}$ conformally equivalent?