The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{2}$

Let $f,g:\mathbb{C}^3\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^3\to \mathbb{C}^3$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
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How to prove that $\mathrm{Proj}\left(B/J\right)$ is isomorphic to $\mathrm{Proj}\left(A/J\right)$ if $I\subset J$?

Let $B$ be a graded ring with positive degrees, and let $I$ and $J$ be homogeneous ideals of $B$. We suppose that there exists $N$ such that $I\cap B_{n}=J\cap B_{n}$ for all $n\ge N$. How to show ...
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Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section ...
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Proving exactness of the conormal sequence

Problem: Let $\phi \colon A \to B$ be a surjective homomorphism of $R$-algebras with kernel $I$. I want to show that the conormal sequence $$ I/I{}^2 \longrightarrow B \otimes_A \Omega_{A/R} ...