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Let $d>1, n>1$ and consider the vector space $V=\mathbb{C}[x_0,\ldots,x_n]_d$. Let $\mathscr A \subseteq \mathbb{P}(V)$ be the set of all forms with the property that their zero locus in $\mathbb{P}^n …

This follows from Bertini's Theorem: see for example here. Note that your variety does not need to be projective. Let $\overline{k}$ be the algebraic closure of $k$. You know that there is an open den …

Let $f:X\to Y$ be a surjective morphism of smooth irreducible varieties over $\mathbb{C}$. Assume further that $Y$ is complete and that every fiber $f^{-1}(y)$ for $y\in Y$ is complete and irreducible …

Let $f: X \to Y$ be a morphism between two quasiprojective, irreducible varieties over the complex numbers, such that the image of $f$ is Zariski dense in $Y$ and there is a Zariski dense subset $U$ ( …

Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' "Algebra …

I am interested in smooth nonedegenerate surfaces $X\subset\mathbb{P}^n$, $n\geq 5$, whose secant variety $\sigma(X)$ has dimension $4$. Clearly, the second Veronese embedding of $\mathbb{P}^2$ is suc …

Is there a characterization of projective varieties $X\subset\mathbb{P}^n$ whose secant variety is a hypersurface of degree $3$? In the case that the secant variety does not have the expected dimensio …

Let $X$ be a projective variety over a field $k$ of characteristic $0$ and let $f: X \to \mathbb{P}^d$ be a $k$-morphism. Can we always find an embedding $i: X \hookrightarrow \mathbb{P}^N$ such that …

Let $f: X \to Y$ be a finite, surjective morphism of smooth, projective, irreducible varieties over $\mathbb{C}$ and let $y \in Y$. Can I find a smooth curve $C \subseteq Y$ with $y \in C$ such tha …

Let $X \subseteq \mathbb{P}^n$ be a projective variety. I would like to have a morphism $f: \tilde{X}\to X \subseteq \mathbb{P}^n$ where $f$ is finite and birational, $f^* \mathcal{O}_{\mathbb{P}^n}(1 …

Let $R \hookrightarrow S$ be a finite extension of noetherian rings. Let $I \subseteq S$ be an $R$-submodule of $S$. Are there any sufficient criteria on $I$ such that it is in fact an ideal of $S$? M …

Let $X\subset\mathbb{P}^n$ be a smooth nondegenerate (i.e. not contained in any hyperplane) curve over $\mathbb{C}$. Is it possible that every collection of $n-3$ points on $X$ lies on a $n-1$-secant …

Let $X \subseteq \mathbb{P}_{\mathbb{C}}^n$ be an irreducible, projective, Cohen-Macaulay variety of dimension $k$. Let $L \subseteq \mathbb{P}_{\mathbb{C}}^n$ be a linear space of dimension $n-k-1$ t …

Let $K$ be a field. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^n_K$ whose scheme theoretic support is a reduced, closed subscheme $X \subseteq \mathbb{P}^n_K$ of dimension $k$. Let $X_1, \ld …

Let $I_X\subseteq \mathbb{C}[x_0,\ldots,x_n]$ be the homogeneous vanishing ideal of a set $X$ of $s$ points in $\mathbb{P}^2$. Let $d_1$ resp. $d_2$ denote the minimal resp. maximal degree of elements …