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Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $\textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian …

Let $A$ be an abelian variety over $\mathbb{C}$ and let $X_m$ the subset of nontrivial $m$-torsion points on $A$. Can we realize $X_m$ as the zero locus of a global section of a suitable vector bundle …

Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, …

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 …

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 $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where $\textrm{Sym}_n(K …

Morally speaking, my question is whether every real smooth projective curve can be deformed in as many real directions as complex directions. Let me make the question precise. Let $H$ be the Hilbert s …

Let $S=K[x,y,z]$. Consider $f\in S_{2d}$ a ternary form of even degree and let $I_f$ be the associated Gorenstein ideal, i.e., all polynomials $G$ such that $G(\partial_x,\partial_y,\partial_z)f=0$. A …

Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) f …

Let $f_1,\ldots,f_r\in\mathbb{C}[x,y]$ and consider the subalgebras $A_1,\ldots,A_r$ of $\mathbb{C}[x,y]$ that are generated by $f_1,\ldots,f_r$, i.e., $A_i=\mathbb{C}[f_i]$. Using some dimension esti …

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 $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron …

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 …