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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Question regarding Geometric meaning of Noether normalization theorem for projective varieties

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states: Let $K$ be an algebraically closed field, $V\subset \mathbb{P}^n(K)$ a variety of ...
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Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in ...
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Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
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Number of $\mathbb{F}_q$-rational points on a smooth variety

From the proof of Weil's conjectures it follows that $|q^k - \# X(\mathbb{F}_{q^k})| = O(q^{k(n - \frac{1}{2})})$, where $X$ is a smooth variety over $\mathbb{F}_q$ and $n = \dim X$ (see for example ...
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Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb ...