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$\DeclareMathOperator\Gr{Gr}$ Let $\Gr(k,n)$ be the set of $k$-dimensional subspaces in affine space $\mathbb{A}^n$ over an algebraically closed field. If $U\subseteq (\mathbb{A}^n)^{\times k}$ is an …

Let $\mathbb{F}$ be an algebraically closed field of characteristic zero, let $k\leq n$ be positive integers, and let $V=\mathbb{F}^n$. One can view the Grassmannian $G(k,V)$ of $k$-planes in $V$ as a …

$\DeclareMathOperator\Gr{Gr}$Consider $\mathbb{C}^n$ endowed with the Hermitian inner product $\langle u,v\rangle=u^*v$, and let $U \subseteq \Gr(k,\mathbb{C}^n)$ be a Zariski open dense subset of the …

I borrow notation from this answer. Let $X,Y\subseteq\mathbb{P}^{n}$ be two irreducible varieties over an algebraically closed field $k$. Consider $$ S^{0}_{X,Y}:=\{(x,y,z)\in X\times Y\times \mathbb{ …

Let $X \subseteq \mathbb{C}^n$ be an irreducible algebraic set that forms a cone, and let $I=I(X) \subseteq \mathbb{C}[x_1,...,x_n]$. Let $m < n$ and $k\leq m$ be positive integers. Is it true that fo …

$\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Seg{Seg}$I apologize in advance for my naïve understandi …

Let $V$ be a $\mathbb{C}$-vector space of dimension $N$, let $d$ be a positive integer, let $l \leq N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom …

$\newcommand{\complex}{\mathbb{C}}$ Let $Seg \subseteq \complex^M \otimes \complex^N$ be the set of elements of the form $v \otimes w$. It is well-known that a general linear subspace of dimension $(M …

Let $\mathbb{k}$ be an algebraically closed field. The symplectic algebraic group is given by $$ \text{Sp}(2n,\mathbb{k})=\{M\in\text{Mat}_{2n}(\mathbb{k})\mid J=M^TJ M\}\quad\text{where}\quad J=\beg …

Let $\mathbb{C}^n$ be the $n$-dimensional complex vector space endowed with the standard Hermitian inner product, let $X \subseteq \mathbb{C}^n$ be an algebraic set that forms a cone, and let $1>\epsi …

Let $V$ be a $\mathbb{C}$-vector space of dimension $N \geq 2$, let $d$ be a positive integer, let $l < N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\b …

Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two q …

Well, actually, what are the dimensions of the following two subvarieties of the Grassmannian. Let $N$ be a positive integer. Let $V \subseteq \mathbb{C}^N$ be a linear subspace of dimension $N-k$ fo …

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous …

$\newcommand{\proj}{\mathbb{P}}\newcommand{\complex}{\mathbb{C}}\newcommand{\ip}[2]{\langle #1 , #2\rangle}\newcommand{\abs}[1]{\lvert #1 \rvert}$There is a natural bijection $\phi: \proj(\complex^n)\ …