# Questions tagged [projective-geometry]

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### Jumping conics in Grassmannians

Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...
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### Moving general fibers of a fibration

Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth. My ...
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### Geometrical meaning of a question from Marden

Let $T \in SL(2,\mathbb{C})$ be a normalised Möbius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$. The above is an exercise from Outer Circles by Marden (ex. ...
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### Irreducibility of the base and of the general fiber

Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible. Does there exists an irreducible component $X'$ of ...
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### Is the projective symmetry group of a polytope more general than its linear symmetry group?

Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups: \begin{align} \DeclareMathOperator{\Aut}{Aut} \...
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### Moduli spaces and conic bundles

The moduli space $A_2(1,8)^{\operatorname{lev}}$ of $(1,8)$-polarized abelian surfaces with canonical level structure has a structure of conic bundle over $\mathbb{P}^2$ with a curve of degree $4$ as ...
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### Canonical class & ring of projective space $\mathbb{P}^n$ in differential geometry

David Mumford remarks in his book Algebraic Geometry I, Complex Projective Varieties on page 109 that the fact that the canonical ring $\oplus_{k=0}^{\infty} \Omega_{k, \mathbb{P}^n}$ of projective ...
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### Degenerations of hyperelliptic coverings

Take six distinct points $p_1,\dots,p_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p_1,\dots,p_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus ...
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### Smallest subset in $P^2 \mathbf F_q$ which cannot be disjointed from itself by a homography

Let $q$ be a power of a prime and $S \subseteq \mathrm P^2 \mathbf F^q$ such that $$\forall g \in \operatorname{PGL}(3,q), gS \cap S \neq \emptyset.$$ Can it be that $\vert S \vert < 1+q$ ? (I ...
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### Determinantal representation of joins

Let $X^n = Z(I_X)\subset\mathbb{P}^N$ be an $n$-dimensional irreducible and non degenerate variety. Consider a linear subspace $H = Z(I_H)\subset\mathbb{P}^N$ of dimension $N-n-2$ disjoint from $X$. ...
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### Set theoretic equation for Veronese varieties

Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e.,...