The projective-geometry tag has no usage guidance.

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134 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

125 views

### Degree of quasi-projective variety [closed]

Why we cannot define the degree of a quasi-projective $k$-variety ($k=\bar k$) $X$ for a given embedding $X\subset \mathbb P^n_k$ ?
If we take any compactification $\bar X$ of $X$, $\bar X\backslash ...

**0**

votes

**0**answers

88 views

### Quadrics cutting out a polygon

Let $l_1,l_2,l_3,l_4,l_5\subset \mathbb P^4_k$ be distinct lines such that $|l_i\cap l_{i+1}|=1$ for all $i\ mod\ 5$ and $l_i\cap l_j\neq \emptyset\iff j=i+1\ mod\ 5$ (so that $\cup_{i=1}^5l_i$ is a ...

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**0**answers

53 views

### Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...

**4**

votes

**3**answers

341 views

### Automorphism group of a variety

Suppose $X$ is a (quasi-projective) variety over a field $k$, and let $\mathbb{P}^n(k)$ be the ambient projective space.
When can one decide that the automorphism group of $X$ is induced by a ...

**5**

votes

**2**answers

273 views

### Automorphisms and infinitesimal deformations of a smooth complete intersection

Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$.
Is it known when $Aut(X)$ is finite ?
Does there exist a formula for the dimension of the tangent space to the ...

**3**

votes

**0**answers

86 views

### Generalizing von Staudt's synthetic construction of the complex numbers

Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...

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votes

**1**answer

156 views

### Sections of the conormal bundle

Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and ...

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votes

**4**answers

484 views

### Synthetic projective lines

The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and ...

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vote

**2**answers

284 views

### Mapping multivariate polynomial inequalities system to subspace

What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution.
I have a set of linear equations, e.g.:
\begin{align}
d_1 &= L_1 - ...

**2**

votes

**1**answer

93 views

### Sections of a sheaf of differentials on a weighted complete intersection

Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then ...

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votes

**7**answers

7k 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 ...

**3**

votes

**1**answer

288 views

### Ramification divisor on curves in weighted projective space

I was hesitant about posting this question here, but since it deals with a partially unanswered question already on this site I figured that this would be the best place for it. I apologise in advance ...

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votes

**3**answers

428 views

### On well-formedness of weighted projective spaces and a Hurwitz theorem calculation

This question has two parts: A calculation that is giving me a lot of troubles, and a theoretical one on weighted projective spaces.
1) I want to find the genus of the curve $C_7 \subset ...

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votes

**2**answers

120 views

### Quartic symmetroids and 10-points sets

A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid ...

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votes

**1**answer

148 views

### Self intersection and deformations

Suppose I have a smooth projective surface $S$ and a curve $C\subset S$. The self-intersection of $C$ is by definition the degree of the restriction to $C$ of the normal bundle of $C$ inside $S$.
By ...

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votes

**1**answer

238 views

### A question on young persons guide to canonical singularities

In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in ...

**4**

votes

**0**answers

644 views

### What is projective duality from modern point of view ? (correspondence ? Fourier on D-mod ? Aut of D(Coh) ?)

Consider vector space $V$ and its dual $V^*$ then to any line subspace in $V$ one can correspond its kernel in $V^*$ which is hyperplane.
Projective duality states that this correspondence satisfies ...

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votes

**2**answers

265 views

### A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically.
A specific construction of a set of ...

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**2**answers

332 views

### Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of ...

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**0**answers

35 views

### cross sections of semialgebric sets

Let n be bigger than two, and let A be a subset of the n-dimensional euclidean space. Suppose that the intersection of A with any (n-1)-dimensional affine hyperplane is semialgebraic. Can one conclude ...

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**0**answers

90 views

### When is an intersection of quadrics empty

Is there an easy way to tell when the intersection of a set of quadrics (in complex projective space) is empty? Assume you know the quadrics explicitly.

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votes

**3**answers

280 views

### Existence of a morphism between two toric varieties

Does there exist a morphism between the blow-up of $\mathbb{P}^3$ in four general points and $\mathbb{P}^1\times\mathbb{P}^1$? If not why?

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**0**answers

69 views

### Intersection of affine subspaces with specific Veronese variety

Consider the mapping
$$ \Psi: \mathbb R^2 \times \mathbb R^2 \to \mathbb R^5, \\
\Psi(x^{(1)},x^{(2)}) = \begin{pmatrix} x_1^{(1)} + x_1^{(2)} \\ x_2^{(1)} + x_2^{(2)} \\ (x_1^{(1)})^2 + ...

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votes

**10**answers

4k views

### What is the exact statement of “there are 27 lines on a cubic”?

I think there was a theorem, like
every cubic hypersurface in $\mathbb P^3$ has 27 lines on it.
What is the exact statement and details?

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votes

**1**answer

176 views

### dimension of singular locus and complete intersection of a hypersurface

Let $X$ be a reduced projective hypersurface over a field $k$, which is defined by the homogeneous equation $f(T_0,\ldots,T_n)=0$. If the dimension of the singular locus of $X$ is $s$, $0\leq s\leq ...

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votes

**1**answer

159 views

### Equivariant Almost Complex Structures on the Full Flag Manifolds

On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the ...

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votes

**1**answer

190 views

### Do general sheaves on P^2 have cohomology governed by their Euler characteristic?

Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$.
If $\xi$ has Euler characteristic 0, then apparently there is ...

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votes

**1**answer

191 views

### GIT quotients and automorphisms

Let $X$ be a smooth projective variety. Then we have an exact sequence:
$$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$
where $Aut^{o}(X)$ and $H$ are respectively the connected ...

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votes

**1**answer

150 views

### Varieties parametrizing skew-symmetric matrices

Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices.
Let $M\in\mathbb{P}(\bigwedge^2V)$, for any ...

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**1**answer

155 views

### How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...

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votes

**2**answers

550 views

### Can the projective line be provided with a ring structure?

A definition of multiplication on the projective $1$-points $(a:b)$ of $P_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the ...

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votes

**2**answers

215 views

### Fibrations on blow-ups of $\mathbb{P}^2$

Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$.
Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...

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votes

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321 views

### References on quaternionic geometry

Is there any analog, in the quaternionic setting, of Kahler potentials?
In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...

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votes

**1**answer

220 views

### Automorphisms of del Pezzo surfaces

Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p_1,p_2,p_3$.
Is it true that its automorphism group is ...

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**2**answers

147 views

### Transformations that leave the Plucker embedding of G(2,4) invariant

I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by ...

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votes

**1**answer

375 views

### What criteria are to determine if two projective varieties are projectively equivalent?

A projective transformation is a morphism of $P^n$ to $P^n$, for some $n$, determined by an
$(n + 1) \times (n + 1)$ invertible matrix $A$ in the obvious way. The sets $Q$, $R$ are projectively ...

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votes

**1**answer

348 views

### Generalized geometries

Let $S$ be a non-empty set. A geometry of type $n$ for $n\geq 1$
on $S$ (consisting of at least $n$ elements) is a set ${\mathfrak P}\subseteq
{\mathcal P}(S)$ such that
all members of $\mathfrak ...

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votes

**2**answers

335 views

### Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...

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votes

**1**answer

209 views

### Infinitesimal deformations of a singular projective surface

Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities.
Assume that both the singularities in $x_1$ and in $x_2$ admit a ...

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**0**answers

127 views

### Independent Generic Curves in the Projective Plane

I'm trying to read a paper by Masayoshi Nagata
(available here)
where he gives a counter-example to Hilbert's fourteenth prolem
and I've run into some trouble understanding the terminology he's using.
...

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vote

**1**answer

155 views

### Morphisms contracting a family of curves

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve.
...

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votes

**3**answers

808 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 ...

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**1**answer

214 views

### Rational curves in projective spaces

Let $X\subset(\mathbb{P}^{N})^n$ be the variety defined as follows: $(p_1,...,p_n)\in (\mathbb{P}^{N})^n$ such that there exists a rational curve $C$ of degree $d$ with $p_1,...,p_n\in C$.
Is there a ...

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vote

**1**answer

115 views

### Schubert Polynomials for Complex Projective Space

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific ...

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**0**answers

74 views

### Rational curves through a fixed number of points

Let us fix two positive integers $d$, and $N$. Can we determine a third integer $n$ such that given $n$ general points $p_1,...,p_n\in\mathbb{P}^N$ there exists a unique rational curve of degree $d$ ...

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212 views

### Singularities of Pfaffian hypersurfaces

Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am ...

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230 views

### Secant varieties of curves in $\mathbb{P}^4$

My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...

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votes

**1**answer

249 views

### Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory.
The facts are stated as follows:
Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally ...

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votes

**2**answers

326 views

### Reference on the Veblen-Young characterization of projective spaces

Can someone point me to a modern treatment of the Veblen-Young characterization of projective spaces of dimension greater than $2$ as $P(V)$ for some vector space $V$?
[Added: see here for a ...