Questions tagged [projective-geometry]

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Blowing-up a non reduced fiber

Let $X\rightarrow \mathbb{P}^2$ be a smooth conic bundle with a non reduced fiber $F$, and $\widetilde{X}$ the blow-up of $X$ along $F$ with exceptional divisor $F\times\mathbb{P}^1$. I expect $\...
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  • 281
3 votes
0 answers
51 views

Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...
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  • 3,097
1 vote
1 answer
53 views

Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$

Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...
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2 votes
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52 views

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 ...
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  • 3,097
1 vote
0 answers
70 views

$0$-dimensional intersection in weighted projective space

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(...
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  • 281
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 ...
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  • 355
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. ...
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  • 355
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 $...
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  • 765
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 ...
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  • 765
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 $L\subseteq\mathbb{R}^2$ ...
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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 ...
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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....
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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 ...
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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 ...
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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 ...
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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 $k\geq d$ ...
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  • 874
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 ...
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  • 351
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 $(-K_X)^2$ ...
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  • 383
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, ...
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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+...
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0 votes
1 answer
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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4 votes
1 answer
237 views

Del Pezzo surfaces of degree four and complete intersections of two quadrics

Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$. Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
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  • 387
1 vote
0 answers
32 views

Projective equivalence for quadrics and their image

Let $\mathbb{K}$ be the complex or real field and $(S_n,\mathcal{K})$ a $\mathbb{K}-$projective space of dimension $n\in\mathbb{N}^*$, where $\mathcal{K}$ is the collection of bijections $\kappa: S\to\...
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  • 111
6 votes
2 answers
474 views

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 ...
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  • 281
1 vote
1 answer
141 views

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 ...
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  • 387
1 vote
0 answers
30 views

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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6 votes
1 answer
184 views

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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  • 1,051
2 votes
1 answer
78 views

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 ...
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4 votes
0 answers
253 views

How many ways are there to characterise $\mathbb{P}^n$?

Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\...
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  • 813
5 votes
1 answer
186 views

Lines on quadric surfaces

Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
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  • 387
2 votes
2 answers
325 views

Embedding of a blow-up

In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H_1,H_2$ of type $(2,1)$ and set $Y = X\cap H_1\cap H_2$. Let $Z$ be the blow-up of $X$ ...
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  • 351
5 votes
2 answers
494 views

Birational geometry over finite fields

I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
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3 votes
0 answers
81 views

Clarifications involving automorphisms of projective planes and lines?

I have been learning some classical projective geometry recently and I am hoping to gain some clarity regarding various different automorphism groups. There are three different levels of generality ...
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  • 383
4 votes
1 answer
115 views

Embedding quadric bundles

Let $\pi:X\rightarrow W$ be a morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric, and let $r$ be the dimension of the fibers of $\pi$. Does there always ...
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  • 387
2 votes
0 answers
71 views

Which projective planes are connected manifolds?

Inspired by a nice recent MO question, I thought I would ask a similar one: which projective planes $P$ can be given a topology and structure of a smooth connected manifold where the lines form smooth ...
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  • 5,201
9 votes
1 answer
339 views

p-adic analogue of octonions

There are the complex p-adic numbers. But what is the p-adic analogue of the Cayley–Dickson construction? Or more important: What is the p-adic analogue of the octonions? It would be nice if the (unit)...
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  • 93
0 votes
0 answers
104 views

Problem in calculating the global sections of $\mathcal{O}_{\mathbb{P}^3}(d)\otimes \mathcal{I}_Z$

This is an additional question to the one I posed in Equivalence of sequences of blowups of $\mathbb{P}^3$ Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the ...
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  • 1,167
2 votes
1 answer
184 views

Equivalence of sequences of blowups of $\mathbb{P}^3$

Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...
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  • 1,167
2 votes
1 answer
132 views

Is a line associated with antipodal points (the fact, it is the generalization of Simson line) known?

First time, I found a line associated with antipodal points, detail: Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ ...
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1 vote
0 answers
61 views

Generalized polynomials and a $4$-point cross-ratio function on the non-degenerate complex $4$-quadric

The Grassmannian $Gr_1(\mathbb{C}^2)$ is another name for $\mathbb{P}^1$. If one endows $\mathbb{C}^2$ with a complex symplectic form, or if one prefers (since this will allow us to generalize in ...
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  • 3,594
4 votes
1 answer
271 views

Cohomology of divisors on Hirzebruch surfaces

Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...
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3 votes
1 answer
165 views

Mori cone of Picard rank two varieties

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ is an isomorphism, where $i:S\...
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  • 355
2 votes
1 answer
277 views

Divisors on projective bundles

Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$. If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
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  • 355
0 votes
1 answer
228 views

Coefficients of elliptic curves over function fields

Consider the projective plane $\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$. Take the point $p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\...
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  • 387
2 votes
1 answer
170 views

Blow-ups of surfaces over a field

Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure. Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
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5 votes
2 answers
412 views

Divisors whose restriction is big

Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$. ...
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2 votes
1 answer
177 views

Trivial rational solution of a system of hyperplanes

Let us consider a vector space $ V $ over $ \mathbb{Q} $ of dim $6$. We denote all the two dimensional subspace in $ V $ by $ G(2,6) $ (The Grassmanian variety). One can define a map $ p $ from $ G(2,...
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  • 693
7 votes
2 answers
280 views

Nef divisors on surfaces

Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties: $...
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  • 355
2 votes
1 answer
157 views

Curves in conic bundles

Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...
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  • 29
4 votes
0 answers
95 views

Projective planes over algebraically closed fields

Suppose I am given a projective plane $P \cong \mathbb{P}^2(k)$ over a (commutative) field $k$. With "projective plane," I mean the point-line geometry (and not, for instance, the scheme): $...
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