# Questions tagged [projective-geometry]

The projective-geometry tag has no usage guidance.

**13**

votes

**1**answer

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### Projective-invariant differential operator

This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...

**-1**

votes

**0**answers

179 views

### On an exercise in section 4 of Chapter I from Hartshorne's Algebraic Geometry

It is about exercise 4.9:
Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$ with $n\geq r+2$. Show that for suitable choice of $P \notin X$ and a linear $\mathbb{P}^{n-1}\subseteq \...

**2**

votes

**0**answers

126 views

### A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...

**4**

votes

**1**answer

248 views

### Is it a new method to construction of a conic, how can prove?

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:
Let $L_1, L_2$ be two parallel lines, ...

**2**

votes

**1**answer

145 views

### Linear subspaces in quadric hypersurfaces

Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.
Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,...

**2**

votes

**0**answers

61 views

### Projectively flat Weyl connection on closed higher genus surface

A Weyl connection on a smooth $n$-manifold $M$ is a torsion-free connection $\nabla$ on its tangent bundle that preserves some conformal structure $[g]$ on $M$. By this I mean that its parallel ...

**2**

votes

**0**answers

145 views

### Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space:
$$\Bbb E/S_n \cong \Bbb P^{n-1}. $$
This is related to a 4d SU(N) Yang-...

**13**

votes

**2**answers

491 views

### Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

I've got ten (projective) planes in projective 3-space:
\begin{align}
&x=0\\
&z=0\\
&t=0\\
&x+y=0\\
&x-y=0\\
&z+t=0\\
&x-y-z=0\\
&x+y+z=0\\
&x-y+t=0\\
&x+y-t=0
...

**2**

votes

**0**answers

38 views

### Is the finite projective plane stable as an extremal set system?

Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...

**3**

votes

**0**answers

72 views

### Are there local invariants for smooth planes?

A smooth plane is a smooth double fibration
$$
\mathbb{RP}^2 \overset{\pi_1}{\longleftarrow} PT\mathbb{RP}^2 \overset{\pi_2}{\longrightarrow} \mathbb{RP}^2
$$
where the system of curves $\pi_1(\pi_2^{...

**3**

votes

**1**answer

84 views

### Osculating spaces of intersection of two varieties

Let $Z = X\cap Y\subset\mathbb{C}^N$ be a manifold given as the intersection of two manifolds $X,Y$ intersecting transversally along $Z$. Let $T_p^kX,T_p^kY,T_p^kZ$ be the $k$-osculating spaces at $p\...

**2**

votes

**0**answers

62 views

### Boundary at infinity for projective manifolds?

Given a compact real projective manifold, is there something "like" a boundary at infinity for its universal cover?
In the case of compact projective manifolds obtained from divisible convex sets (...

**22**

votes

**6**answers

1k views

### About the definition of E8, and Rosenfeld's “Geometry of Lie groups”

I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...

**5**

votes

**1**answer

205 views

### Intersections in $\mathbb{P}^1\times\mathbb{P}^1$

Let $F$ be an algebraically closed field and $\mathbb{P}^1$ the projective line over $F$. Suppose $V_1, V_2$ are two 1-dimensional subvarieties of the 2-dimensional variety $\mathbb{P}^1\times\mathbb{...

**0**

votes

**0**answers

108 views

### Derived Category of the Fano 4fold variety of lines

Let $X\subset P^5$ be a smooth cubic fourfold. It is well known that its variety of lines $F(X)$ is a smooth fourfold Fano variety. Hence its derived category should have a semi-orthogonal ...

**1**

vote

**0**answers

170 views

### Dimension projectivised tangent space equal dimension variety+1

I'm reading some lecture notes (unfortunatey in italian, https://me.unitn.it/system/files/Bernardi%20Alessandra/tesi_teroni_1.pdf , page 5), where there's this statement (without proof):
Suppose we ...

**6**

votes

**1**answer

258 views

### Generalized projective spaces, spheres, and exotic spheres [closed]

I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres:
The real projective space
$\mathbb{RP}^1 \simeq S^1,$
is ...

**5**

votes

**0**answers

211 views

### $N$-$th$ closed chain of six circles

Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...

**2**

votes

**1**answer

181 views

### Yiu's equilateral triangle-triplet points

In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. In more than ...

**14**

votes

**3**answers

871 views

### An ellipse through 12 points related to Golden ratio

I am looking for a proof of the problem as follows:
Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$...

**2**

votes

**0**answers

146 views

### Extension of a rational section of a projective bundle

Let us assume that we work over the complex field and let $X$ be a smooth projective variety and $\pi: P \to X$ a projective bundle (i.e. a fibration in projective spaces of constant dimension). Let $...

**3**

votes

**2**answers

200 views

### Stability and complete types (in Model Theory)

I read the following statement in these slides of Saharon Shelah:
"$K$ is stable iff for every $M \in K$ there are only "few" complete types
over $M$." About the notation: here $K$ consists of all ...

**2**

votes

**0**answers

99 views

### Intersection number of two projective curves using the resultant and tangent lines

For my thesis, I'm working on the intersection of projective plane curves over $\mathbb{C}$. We define the intersection number of projective plane curves (see for example Gibson - Elementary Geometry ...

**0**

votes

**0**answers

67 views

### Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$.
We call $\mathcal{O}(q)$ the usual twisted line bundle.
Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by
$$[x_0,\ldots, x_n]\...

**8**

votes

**2**answers

205 views

### Linear sections of Segre varieties and rational normal scrolls

In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\...

**0**

votes

**0**answers

95 views

### Non-degenerate varieties in projective space

Let $X\subseteq \mathbb{P}^n(K)$, where $K$ is an algebraic closed field, be a projective variety. $X$ is called non-degenerate if $X$ is not contained in any hyperplane. For a given variety $X$, is ...

**4**

votes

**1**answer

104 views

### Uniqueness of Mukai presentation of canonical model in genus 6

In his 92 paper, Mukai showed that a general genus $6$ curve may be represented in $\mathbb{P}^9$ as the intersection of the Grassmannian $G(2,5)$ (under the Plucker embedding), a plane $H\cong \...

**5**

votes

**1**answer

130 views

### Rational map given by pfaffians

Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf_0(Z):\dots:pf_n(Z)]$ where the $pf_i(Z)$ are the principal ...

**2**

votes

**1**answer

133 views

### Very symmetric quadrangle in $\Bbb CP^2$

Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$?
Clearly the analogous ...

**5**

votes

**1**answer

181 views

### Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$

The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics.
Does anyone know a reference where these 10 quadratic equations are written down ...

**2**

votes

**1**answer

202 views

### Closure of quasi projective scheme

If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a smooth $S$-scheme with $X$ an open subscheme?

**3**

votes

**0**answers

165 views

### Picard group of quasi-projective varieties

Let $X$ be a smooth open sub-variety of a projective, not necessarily smooth, variety $X'$, defined over a finite field.
Is $\text{Pic}(X)$ a finitely generated abelian group?
I'm tempted to just ...

**5**

votes

**1**answer

176 views

### Intermediate moduli spaces of stable maps

In the following paper:
A. Mustata, M. A. Mustata, "Intermediate moduli spaces of stable maps", Invent. math. 167, 47–90 (2007)
the authors introduced a variation on moduli spaces of stable maps ...

**6**

votes

**2**answers

233 views

### Rational normal curves and tangent lines

Let $C,\Gamma\subset\mathbb{P}^n$ be degree $n$ rational normal curves in $\mathbb{P}^n$, such that for any $p\in C$ the tangent line $T_pC$ of $C$ at $p$ is tangent to $\Gamma$ as well. This means ...

**2**

votes

**0**answers

113 views

### On the classification of spherical varieties

Let $G$ be a connected reductive algebraic group, for instance take $G = SL_n$. Does there is a classification of the $\mathbb{Q}$-factorial normal projective varieties with given dimension and Picard ...

**2**

votes

**0**answers

115 views

### Flip of moduli space of stable maps

Let $\overline{M}_{0,2}(G(3,7),4)$ be the moduli space of $2$-pointed degree $4$ stable maps to the Grassmannian of $3$-planes in $\mathbb{P}^7$. Consider the divisor $\Delta$ whose general point is a ...

**2**

votes

**1**answer

185 views

### Compactifications of spaces of morphisms

Let us denote by $Mor_3(\mathbb{P}^1,\mathbb{P}^3)$ the spaces of degree three morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^3$,
$$f(x_0,x_1)=[f_0(x_0,x_1):f_1(x_0,x_1):f_2(x_0,x_1):f_3(x_0,x_1)]$$
...

**2**

votes

**0**answers

127 views

### Reference for the Koszul--Malgrange Theorem

The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...

**4**

votes

**1**answer

193 views

### Ring of sections and normalization

Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational.
Now, let $X(D)...

**3**

votes

**1**answer

116 views

### Flipping and flipped loci

Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ ...

**2**

votes

**0**answers

158 views

### On a class of loci in Chow varieties

Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$.
For $0\le p\le d$,...

**6**

votes

**1**answer

196 views

### Where do the (Akizuki)-Nakano Identities First Appear

The answers to this M.O. question give a history of the Kaehler identities. The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira ...

**3**

votes

**1**answer

152 views

### Curves contracted by a rational map

Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring
$$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$
is finitely generated and ...

**7**

votes

**2**answers

315 views

### Infinite projective plane with small edges

Let $\kappa$ be an infinite cardinal. We say $E\subseteq {\cal P}(\kappa)$ is an infinite projective plane on $\kappa$ if
$e_1\neq e_2\in E$ implies $|e_1\cap e_2| = 1$, and
whenever $n\neq m\in \...

**10**

votes

**1**answer

211 views

### G-modules and ideals of secant varieties

Consider the action of $G = SL(n+1)$ on $\mathbb{P}^N$, and embed $\mathbb{P}^n$ in $\mathbb{P}^N$ via the degree two Veronese embedding. Let $V\subset\mathbb{P}^N$ be the corresponding Veronese ...

**5**

votes

**0**answers

370 views

### Variational Hodge Conjecture vs Hodge Conjecture

Motivation.
Let us state the following version of Grothendieck's variational Hodge Conjecture:
Conjecture (VHC). Let $\mathcal{X}\to S$ be a proper smooth map of smooth algebraic varieties over $\...

**15**

votes

**1**answer

296 views

### Birational automorphisms of varieties of Picard number one

Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?

**10**

votes

**1**answer

473 views

### $K_0$-equivalence of varieties

Let $k$ be an algebraically closed field of characteristic zero.
Let $A$ be the $\mathbf{Z}$-subalgebra of the Grothendieck ring of $k$-varieties $K_0(\text{Var}_k)$ generated by classes of semi-...

**6**

votes

**1**answer

152 views

### Blowing-up an ideal generated by squares

Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...

**1**

vote

**1**answer

171 views

### If $A\subset B$, what to say about their $\operatorname{Proj}$?

Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)...