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Let $X$ be a geometrically irreducible affine variety defined over $\mathbb{Q}$ and dimension $1$. Then it is known that $X$ is birational over $\mathbb{C}$ to a smooth projective curve $C$. I was int …

Let $F_i$ be homogeneous forms with $\mathbb{C}$ coefficients in $n$ variables for each $i$. Let $$ T_2 = \{ (\mathbf{x}, \mathbf{y}) \in \mathbb{A}_{\mathbb{C}}^{2n} : F_i(x_1y_1, ..., x_n y_n) =0, …

Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$. I would like to show that $$ \{ \mathbf{x} \in \mathbb{A}^n_{\m …

Let $N$ be the number of degree $d$ monomials in $n$ variables. We can then view each non-zero point in $\mathbb{A}^N_k$ as a degree $d$ homogeneous form, $k$ an algebraically closed field. Let $X$ be …

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial d …

Let $F$ be a homogeneous from in $\mathbb{R}[x_0, .., x_n]$. Then $F$ defines a projective variety $X \subset \mathbb{P}_{\mathbb{C}}^n$. Assume $X$ is smooth. In this case $F=0$ also defines a subman …

Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations $$ A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C $$ $$ B_{1, 1} x_1 y_1 + B_ …

There is a following result in Mumford's red book of schemes (Chapter II Section 8). Here $R$ is a valuation ring with algebraically closed fraction field $k$. Let $Z \subset \mathbb{P}^n_k$ be an i …

In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows. Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that …

Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the fo …

Let $f: X \rightarrow Y$ be a morphism of schemes: $X$, $Y$ are regular schemes, let $Z_1, Z_2$ be two closed regular subschemes of $Y$, let $x \in X \times_Y Z_2$ such that $y = f(x) \in Z_1 \cap Z_2 …

Let $F: \mathbb{R}^{n} \to \mathbb{R}$ be a smooth function. Suppose $B$ be a closed and bounded box. I would like to obtain for fixed $q \in \mathbb{N}$ $$ \# \{ \mathbf{a} \in \mathbb{Z}^n : F(\f …

Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider …

I'm reading this set of notes and I'm trying to understand this passage where they explain how to put a topology on $GL_n(R)$ when $R$ is a topological ring, which I am not completely following. The f …

Let $G$ be a semisimple linear algebraic group. Let $X^*$ be the group of characters and $X_*$ be the group of cocharacters. Then I know that there exists a pairing $\langle\,,\rangle : X^*(G) \times …