The projective-geometry tag has no wiki summary.

**4**

votes

**1**answer

186 views

### Rectifying texture from image

I have a camera matrix $P$ which defines a projective transformation $\mathbb{P}^3 \rightarrow \mathbb{P}^2$. In the former space there is a plane $[ x|\pi^Tx=0 ]$. The image of the plane under $P$ ...

**59**

votes

**6**answers

6k views

### Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$
Here is a somewhat more conceptual ...

**1**

vote

**1**answer

150 views

### pencil of quadrics consisting of singular quadrics

A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if:
(1) quadrics in $l$ have a common singular point; or
(2) quadrics in $l$ contain a common ...

**0**

votes

**0**answers

25 views

### singular locus of $\mathcal{A}_3(2)^{hyp}$

The geometry of the Satake compactification of $\mathcal{A}_2(2)$ is very well known. It is the singular quartic 3-fold in $P^4$ known as $Igusa\ quartic$. I am looking for references (or ...

**14**

votes

**6**answers

2k views

### Maps to projective space determined by a line bundle

The following should be pretty standard for any algebraic geometer.
Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for ...

**3**

votes

**0**answers

97 views

### Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors.
Have the possible universal covering spaces of $U$ been classified?
Do we know when the ...

**4**

votes

**1**answer

246 views

### The space of varieties between two given varieties

Let $\mathbf{P} = \mathbf{P}^n(k)$ be the $n$-dimensional projective space over a field $k$, let $A, B$ be projective varieties in $\mathbf{P}$ such that $A \subset B$. Now define
$V(A,B)$ to be the ...

**1**

vote

**1**answer

97 views

### Multiplicity of a variety along a subvariety

Let $X\subset\mathbb{P}^n$ be an hypersurface given by the vanishing of a polynomial $F\in k[x_0,...,x_n]_d$. Let $Y\subset X$ be a subvariety. Then $X$ has multiplicity $m$ along $Y$ if all the ...

**2**

votes

**0**answers

113 views

### A question on resolution of singularities

I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$.
For instance consider a cubic surface ...

**2**

votes

**1**answer

126 views

### Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
...

**13**

votes

**1**answer

484 views

### When is $(q^k-1)/(q-1)$ a perfect square?

Let $q$ be a prime power and $k>1$ a positive integer. For what values of $k$ and $q$ is the number $(q^k-1)/(q-1)$ a perfect square, that is the square of another integer? Is the number of such ...

**2**

votes

**0**answers

80 views

### Degree of join of two varieties

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...

**14**

votes

**1**answer

359 views

### Maps to projective space == line bundles; what do maps to weighted projective space correspond to?

A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective ...

**1**

vote

**1**answer

184 views

### Cremona transformations

Let $f:\mathbb{P}^n_1\dashrightarrow\mathbb{P}^n_2$ be the standard Cremona transformation based on $p_1,...,p_{n+1}\in\mathbb{P}^n_1$ and $q_1,...,q_{n+1}\in\mathbb{P}^n_2$. That is, $f$ is the ...

**2**

votes

**2**answers

231 views

### Schwarzian derivative of a diffeomorphism is zero iff Linear-fractionals?

I have found the following derivation of the Schwarzian derivatives in the book of Ovsienko and Tabachnikov:
For a diffeomorphism $\gamma$ which acts on 4 points $t_1,t_2,t_3,t_4 \in \mathbb{RP}_1$, ...

**9**

votes

**1**answer

374 views

### $6$ points lie on a conic if and only if $ABC$ and $A_0B_0C_0$ are perspective

Let $ABC$ be a triangle with incircle $\omega$. Let $A_0,B_0,C_0$ be points outside $\omega$. The tangents from $A_0$ to $\omega$ intersect $BC$ at $A_1,A_2$. Define $B_1,B_2$ and $C_1,C_2$ similarly. ...

**6**

votes

**3**answers

869 views

### Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an english translation, preferrably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...

**0**

votes

**0**answers

118 views

### classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...

**1**

vote

**1**answer

146 views

### Rational normal curves as set-theoretic complete intersections

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$. It is know that $C$ is a set-theoretic complete intersection and that, if $n\geq 3$, is a not a scheme-theoretic complete ...

**2**

votes

**1**answer

92 views

### dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...

**1**

vote

**0**answers

227 views

### Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for ...

**5**

votes

**1**answer

385 views

### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...

**4**

votes

**2**answers

315 views

### A question about pairs of lines in 3D projective space

Consider a 3-dimensional projective space $X$.
Let $m$ be the smallest number so that there are $m$ pairs of lines
$ \ell_1,\ell'_1$, $ \ell_2,\ell_2'$, ... , $\ell_m, \ell'_m$ in $X$:
a) For ...

**6**

votes

**2**answers

374 views

### Reference for Weighted Projective Stacks

For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on ...

**3**

votes

**5**answers

714 views

### When does a planar ternary ring uniquely coordinitise a projective plane?

From every projective plane a coordinitisation can be constructed on a planar ternary ring, and conversely from every planar ternary ring a projective plane can be constructed. (For background see ...

**6**

votes

**0**answers

97 views

### Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...

**5**

votes

**1**answer

427 views

### Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...

**0**

votes

**0**answers

29 views

### Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function
$f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...

**2**

votes

**1**answer

205 views

### Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...

**2**

votes

**1**answer

89 views

### Point-Hyperplane incidence in finite projective spaces

Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of ...

**2**

votes

**1**answer

298 views

### degree 7 rational curves through ten points in P4

This is a very classical flavoured question, and probabaly it is not difficult. I would like to know the shape of the space of rational degree 7 curves in $P^4$ that pass through 10 fixed points. By ...

**11**

votes

**3**answers

297 views

### (Non-)Existence of curves of low degree on affine and projective varieties

I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...

**3**

votes

**2**answers

676 views

### Ample divisors on blown-up projective space

Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...

**1**

vote

**1**answer

174 views

### Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...

**0**

votes

**0**answers

82 views

### Embed the normalization of a curve in a larger space

I'd like to believe that this problem has a positive answer, but I don't know a nice reference. Actually I've never worked with embedded curves, so I apologize in advance if the question is too silly.
...

**2**

votes

**2**answers

172 views

### Curve of 3-secant lines

Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety ...

**2**

votes

**0**answers

107 views

### semi-orthogonal decompositions and embeddings

This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition
$$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$
Suppose ...

**1**

vote

**2**answers

214 views

### Möbius transformation by 3 points in the Minkowski model

Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...

**1**

vote

**1**answer

93 views

### Non-equivariant vector bundles over complex projective $N$-space

From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist ...

**4**

votes

**3**answers

640 views

### (Second) Chern class of projective space, blown up in a linear subvariety

I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to this question, but I cannot ...

**1**

vote

**1**answer

266 views

### Does every ample divisor “span” a hyperplane?

Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...

**2**

votes

**1**answer

95 views

### minimality/universality of the Springer resolution of a determinantal variety

Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some ...

**2**

votes

**1**answer

93 views

### Flag primitivity of the correlation group of classical projective planes.

We know that the full automorphism group of the $\pi_q = PG(2,q)$ acts imprimitively on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full ...

**4**

votes

**1**answer

169 views

### What is the formula for the commutative multiplication on CP(infinity)?

There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian ...

**2**

votes

**2**answers

413 views

### Degree of a smooth projective variety

Let $i_1:X \hookrightarrow \mathbb{P}^n$ and $i_2:Y \hookrightarrow \mathbb{P}^N$ be two projective schemes.
Let $f:X \to Y$ be a surjective projective morphism between smooth projective varieties ...

**5**

votes

**0**answers

157 views

### Non minimal K3 surfaces as hypersurfaces of weighted projective spaces

I recently learnt that the hypersurface
$$
S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1)
$$
is birational to a K3 surface. This is surprising because the surface is quasi-smooth, ...

**0**

votes

**1**answer

216 views

### singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...

**3**

votes

**1**answer

139 views

### Dimension of the global sections of the Serre twisting sheaf on a curve

Let, $C$ be a projective curve (not necessarily reduced), $i:C \to \mathbb{P}^n$ be a closed immersion. Does there exist a bound on/geometric interpretation of the dimension of ...

**4**

votes

**1**answer

195 views

### Extension of linear system

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on ...

**7**

votes

**2**answers

279 views

### Minor theorems of Pappus and Desargues in “old school” geometry?

My question concerns the dependence relations between the minor theorem of Pappus which, following Heyting, I will denote by $P_9$, and (one of the) minor theorems of Desargues, $D_9$.
$P_9$ states ...