# Questions tagged [projective-geometry]

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504
questions

**5**

votes

**1**answer

280 views

### Golden ratio as a property of conic section (is it known?)

I am looking for a proof of a discovery as follows:
Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...

**1**

vote

**1**answer

233 views

### Thirteen-point conic and four-point line, are they new?

We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...

**13**

votes

**1**answer

1k views

### Is it a new discovery on conic section?

I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...

**3**

votes

**0**answers

90 views

### Nodes of rational plane sextic curves

According to the genus formula for plane curves, a nodal, irreducible rational plane curve $C\subset\mathbb{P}^2$ of degree six must have $10$ nodes.
Now, if we take $10$ general points in $\mathbb{P}^...

**3**

votes

**1**answer

101 views

### Projective invariants of the plane and cross ratio

I am looking for a reference for the following admittedly imprecise statement:
Any projective invariant of n points in the projective plane may be
expressed as a function of well-chosen cross-ratios.
...

**9**

votes

**0**answers

123 views

### Projective planes over non-division rings

Is there a "right" notion of a projective plane over a general (unital, non-division) ring?
Let me explain what type of object I am looking for. Let $R$ be an arbitrary (not necessarily ...

**4**

votes

**0**answers

113 views

### Sections of fibrations of Kodaira dimension zero

Let $X$ be a projective variety with a morphism $f:X\rightarrow \mathbb{P}^1$, and let $F$ be a general fiber of $f$.
Assume that $F$ in turn has a fibration $g_{F}:F\rightarrow S$ with rational ...

**3**

votes

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84 views

### Inverse flip of terminal 3-fold

Let $X$ be a terminal $3$-fold over $\mathbb{C}$ and $C\subset X$ a rational curve with normal bundle $N_{C/X} = \mathcal{O}_C(-2)\oplus\mathcal{O}_C(-k)$ for some integer $k > 0$. Assume that $-K_{...

**2**

votes

**1**answer

196 views

### Tangent space to spaces of maps

Let $B = \{x_1,\dots,x_{d-2},y_1,\dots,y_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x_1,\dots,x_{d-2}$ to a fixed point $...

**1**

vote

**1**answer

137 views

### A question on linear projection of a smooth projective variety

Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n$. Fix a closed point $x \in X$ and an embedding of $X$ in $\mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $\sigma_d ...

**10**

votes

**1**answer

383 views

### 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 $Gr(1,n)$ ...

**0**

votes

**1**answer

99 views

### 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 ...

**2**

votes

**0**answers

57 views

### 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. ...

**4**

votes

**1**answer

155 views

### 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 ...

**2**

votes

**0**answers

81 views

### 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}
\...

**4**

votes

**1**answer

168 views

### 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 ...

**2**

votes

**0**answers

76 views

### 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 ...

**3**

votes

**1**answer

130 views

### 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 ...

**2**

votes

**1**answer

58 views

### 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 ...

**3**

votes

**0**answers

71 views

### 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 $X$.
...

**10**

votes

**1**answer

180 views

### 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 ...

**3**

votes

**1**answer

102 views

### 3-secant lines of a projective curve

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.,...

**5**

votes

**1**answer

272 views

### Higher order inflection points

Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T_xX$, of $X$ at $x\in X$, intersects $X$ in $x\...

**3**

votes

**1**answer

143 views

### Configuration of points on a plane curve

Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there ...

**2**

votes

**1**answer

134 views

### Picard groups of determinantal varieties

Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...

**2**

votes

**1**answer

202 views

### Help about “Varieties with small Dual Varieties” by L.Ein

I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...

**4**

votes

**1**answer

153 views

### Linear spaces secant to Veronese varieties

The following question makes sense in a more general setting but for sake of simplicity let me stick to a particular case.
Consider the degree three Veronese embedding $V\subset\mathbb{P}^9$ of $\...

**5**

votes

**1**answer

184 views

### Software computing dimension and degree

Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ ...

**1**

vote

**0**answers

67 views

### Strict transforms of higher codimension subvarieties

The following question could be posed in a more general context but for simplicity I will stick to a particular case.
Let $X\subset\mathbb{P}^9$ be a smooth variety of degree $d$ and dimension $6$. ...

**2**

votes

**0**answers

131 views

### Quadrics tangent to lines

I think that the following must be a basic question in enumerative geometry.
Take a line $L\subset\mathbb{P}^3$. The quadric surfaces in $\mathbb{P}^3$ that are tangent to $L$ are parametrized by a ...

**0**

votes

**1**answer

53 views

### Vertices of 2 self-polar triangles lie on conic

I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?

**9**

votes

**2**answers

572 views

### Picard group of a cubic hypersurface

Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...

**4**

votes

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71 views

### Maximal abelian subgroups of the full collineation group $\mathrm{P\Gamma L}_3(q)$

Is there a convenient list of the maximal abelian subgroups of the projective semilinear group $\mathrm{P\Gamma L}_3(K) \cong \mathrm{PGL}_3(K) \rtimes \mathrm{Gal}(K)$ for $K$ a finite field?
This is ...

**2**

votes

**0**answers

89 views

### Divisorial contraction to a non-normal variety

Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...

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vote

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94 views

### Trisecant lines to curves in $\mathbb{P}^3$: a reference request

I'm starting to study the geometry of the locus of trisecant lines to a space curve $C \subset \mathbb{P}^3$. The main references, cited in almost all the papers, regarding the specific calculations ...

**5**

votes

**2**answers

155 views

### Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$

Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}_S(1,1)$.
If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$...

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vote

**0**answers

124 views

### Cubic surface in $\mathbb{P}^3$ singular along a line

Maybe it is a stupid question but I'm not able to find the answer anywhere else.
My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...

**0**

votes

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79 views

### Lines through the origin every pair of which meet at the same angle

This item isn't getting attention, so I'll try it here:
begin quote
The three lines through antipodal pairs of centers of faces of a cube meet each other pairwise at $90^\circ$ angles.
The three lines ...

**4**

votes

**1**answer

139 views

### Smoothness of moduli spaces of stable maps

If $X$ is a projective variety the moduli space of stable maps $\overline{M}_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities.
Can the pairs $(X,\beta)$ such that $\overline{M}_{...

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vote

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105 views

### Kähler fiber space with base and fiber projective

Let $X$ be a Kähler manifold, $Y$ be a projective manifold, if $X$ exits a smooth fibration over $Y$ such that all the fibers are projective manifolds, then is $X$ a projective mannifold?
If we do not ...

**0**

votes

**1**answer

88 views

### Does projective transformation preserve convexity? [closed]

Does projective transformation preserve convexity?
Notice: Ignore the trivial case which projects a convex curve to a straight line.

**5**

votes

**1**answer

213 views

### Isomorphisms of complete intersections

Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...

**3**

votes

**0**answers

64 views

### Infinite-dimensional quasifields

In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and ...

**5**

votes

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85 views

### Divisorial contractions and singularities

I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...

**2**

votes

**1**answer

240 views

### A question on a Macaulay2 computation

I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.
Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...

**9**

votes

**1**answer

504 views

### What is the automorphism group of the projective line minus $n$ points?

$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...

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158 views

### Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$

Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...

**7**

votes

**1**answer

209 views

### Strict transform of a tangent curve under blow-up

$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...

**4**

votes

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127 views

### Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...

**5**

votes

**0**answers

382 views

### When a fibration over a projective manifold is projective?

Let $X$ be a complex manifold, $B$ be a complex projective manifold, consider a smooth fibration $\pi:X\rightarrow B$ such that all the fibers of $\pi$ are simply-connected projective manifolds, then ...