# Questions tagged [projective-geometry]

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### Nonlinear automorphisms of projective spaces and the axiom of choice

Let $k$ be a field and $\mathbf{P}$ a projective space over $k$. If we accept the axiom of choice (AC), then $\mathbf{P}$ has a basis and a dimension $m$, and if $m$ is finite, the automorphism group ...
1 vote
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Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by $$\phi(... 2 votes 1 answer 93 views ### Surfaces with rational double points Let S\rightarrow \mathbb{P}^1 a surface fibered in conics over a field. Assume that S has a single non reduced fiber F with two points of type A_1 on it. Blowing-up the two points and ... 4 votes 1 answer 126 views ### Singularities of surfaces fibered in rational curves Let S be a projective surface with a morphism S\rightarrow\mathbb{P}^1 whose fibers are either smooth \mathbb{P}^1's or the union of two smooth \mathbb{P}^1's intersecting in a point. ... 0 votes 0 answers 97 views ### Projectivization of graded vector spaces Let E=E^0\oplus E^1 be a \mathbb{Z}_2-graded vector space. Is there a somehow graded notion of projectivization$$\mathbb{P}(E)=\mathbb{P}(E^0\oplus E^1)$$maybe in terms of \mathbb{P}(E^0) and ... 1 vote 0 answers 132 views ### Projectivization in the derived category of coherent sheaves Let X be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over X. Does that concept extend to objects in the derived category ... 6 votes 1 answer 352 views ### Functions \mathbb{R}^2\to\mathbb{R}^2 that preserve lines The simplest case of the Fundamental Theorem of Projective Geometry states that, if f: \mathbb{R}^2\to\mathbb{R}^2 is a bijection that preserves lines – in the sense that if L\subseteq\mathbb{R}^2 ... 3 votes 0 answers 29 views ### Anti-flag transitive affine planes Let \mathcal{A} be an axiomatic affine plane. First let \mathcal{A} be finite. Suppose that the automorphism group of \mathcal{A} acts transitively on nonincident point-line pairs (that is, on ... 2 votes 1 answer 128 views ### Question regarding linear system of projective space I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2.... 1 vote 1 answer 86 views ### Number of orbits for abelian group actions Suppose G is an abelian group acting faithfully on two sets, X and Y, of the same size. None of G, X and Y is finite. Now suppose G is the union of abelian groups G_i, where i varies ... 2 votes 0 answers 123 views ### Is the ideal of the Veronese variety V_{d,n} generated by quadrics? Maybe it sounds like a silly question to the experts but I'm not able to find a proper reference in the web. Anyone knows if the ideal I_{d,n} of the Veronese variety V_{d,n} is generated by ... 2 votes 0 answers 64 views ### Anti-flag transitive projective planes Let \Gamma be an axiomatic projective plane, and suppose its automorphism group acts transitively on the anti-flags (the point-line pairs (u,V) such that u is not incident with V). In the ... 0 votes 1 answer 109 views ### A variation on the projective Nullstellensatz Let V be a \mathbb{C}-vector space, and let f_1,\dots,f_n \in S^d(V^*) be homogeneous polynomials of degree d for which V(f_1,\dots, f_n)=\{0\}. Must there exist a positive integer k\geq d ... 7 votes 1 answer 381 views ### Singular curves of genus 1 Let C be an irreducible curve of arithmetic genus 1 over a field k and with a double k-point p\in C. Is C rational over k? If C is a plane cubic the answer is positive since we can ... 5 votes 1 answer 183 views ### Volume of conic bundles Consider a smooth conic bundle X\rightarrow \mathbb{P}^1 with discriminant of degree d (the locus of \mathbb{P}^1 over which the fibers are reducible conics). There is a formula for (-K_X)^2 ... 3 votes 1 answer 187 views ### Contact variety to projective variety is equidimensional I first asked this question at math.stackexchange with no success, so I decided to repost it here. I am reading the paper "Weakly Defective Varieties" by L. Chiantini and C. Ciliberto, ... 3 votes 1 answer 244 views ### Smooth surfaces in positive characteristic Let K = \mathbb{F}_p be a field of positive characteristic p > 0. Consider a surface in \mathbb{A}^3_K of the following form$$ S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+... 135 views

### Hypersurface on $\mathbb{P}^m\times\mathbb{P}^n$ [closed]

I have found a statement (Harris "Algebraic Geometry", p.269) saying that a hypersurface in $\mathbb{P}^m\times\mathbb{P}^n$ can be written as a single equation. I couldn't find the proof. ...
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### Smooth complete intersections

Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...
1 vote
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### Space of rational conics

Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$. Conisider the ...
1 vote
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### Image points to a plane and computing the covariance for a noisy observer

Let's assume a camera in space and an image point in this camera: $t \in \mathbb{R}^3$ is the position of the camera in space. $R \in \mathbb{R}^{3 \times 3}$ is the orientation of the camera in ...
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### Mirror symmetry for K3 fibered Calabi-Yau threefolds

By a K3 fibered Calabi-Yau threefold, I mean a smooth projective threefold $X$ with trivial canonical class and $h^{1,0}(X) =h^{2,0}(X) = 0$ that has a fibration $X \rightarrow \mathbb P^1$ whose ...
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### Linear subspace in quadric hypersurfaces over a field

Let $K$ be a field of characteristic different from two, and $Q\subset\mathbb{P}^{n+1}_K$ an $n$-dimensional smooth quadric hypersurface over $K$. Suppose also that $Q$ has a $K$-point and so $Q$ is ... 253 views

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