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3
votes
0answers
20 views

Intersection multiplicity of limit linear spaces

Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$. Now let $\Lambda_{\xi, q}$, with $\xi \...
0
votes
0answers
48 views

A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic

I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following: Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...
10
votes
2answers
553 views

Are there nonlinear projective spaces?

This is actually a series of questions posed by Guram Berishvili about the structure he calls marao. Everything I am going to write here I took (and messed up) from his home page which is all in ...
0
votes
0answers
48 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. ...
9
votes
1answer
238 views

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

In the book of Garrett 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 ...
0
votes
1answer
95 views

Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$

I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals. The ...
3
votes
0answers
71 views

Approximating formal surfaces by analytic surfaces

Let $C$ be a smooth projective curve contained in a smooth projective $3$-fold (everything defined over $\mathbb C$). Let $\widehat X$ be the formal completion of $X$ along $C$ and suppose there ...
3
votes
1answer
87 views

Smooth curves in Tangent Developables

Let $C\subset\mathbb{P}^n$ be a smooth curve, and let $Y\subseteq\mathbb{P}^n$ be its tangent developable. Given two general points $y_1,y_2\in Y$ does there exist a smooth curve $\Gamma\subset Y$ ...
7
votes
1answer
261 views

Cohomology of tangent sheaf of a singular hypersurface

Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s. Does there exists a formula, perhaps in ...
5
votes
3answers
347 views

Elementary reference for the isometry group of $\mathbb{RP}^2$

Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ". It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an ...
0
votes
2answers
198 views

Does there exist an Affinization or Projectivization process for Varieties?

Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...
3
votes
1answer
93 views

Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points. Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...
14
votes
1answer
334 views

Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected. If there exists ...
1
vote
0answers
47 views

Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$

Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I_n$ be the $n\times n$ identity matrix, and let $A_n$ be the $n\times n$ ...
6
votes
2answers
268 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 ...
4
votes
1answer
265 views

Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note) Consider the Bundle theorem configuration : Points $A_1, A_2, A_3, A_4$ lie on a circle, ...
7
votes
1answer
477 views

A chain of six circles associated with a conic

I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems: A chain of six circles ...
1
vote
0answers
59 views

Quadrics passing through a point of a variety that are parametrized by a quadric

Let $X\subset\mathbb{P}^{N}$ be a $n$-dimensional algebraic variety and let $x\in X$. Let us suppose that $$ \hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\} $$ ...
17
votes
6answers
3k 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
0answers
142 views

Conjecture generalization of Feuerbach theorem and somes another theorems

My question: I am looking for a solution of a conjecture generalization of the Feuerbach theorem in the end of the topic. But I think, I should let you know why I found this conjecture. I thank to You ...
1
vote
1answer
154 views

Ideal of rational curve in projective space

Let $C_d\subset \mathbb P^d$ be a rational curve of degree $d>2$ ($C_d$ can be reducible) and $n\geq d$. Do we always have $h^0(I_{C_d}(n))=\binom {n+d}{d} - nd-1$?
2
votes
2answers
237 views

Curves in homogeneous varieties

Let $C$ be a curve in a projective homogeneous variety $X$. Fixed a general point $x$ in $X$, does there exist a curve $V$ in $X$ passing through $x$ and such that $C$ and $V$ have the same homology ...
38
votes
11answers
3k views

What is the Cayley projective plane?

One can build a projective plane from R^n, C^n and H^n and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as OP^2, the Cayley projective plane....
4
votes
4answers
266 views

Hilbert metric for the Euclidean plane, a reference?

The Euclidean plane can be seen as an affine chart of the projective plane, together with a scalar product on the line at infinity, so that the absolute consists of two conjugate complex points at ...
0
votes
0answers
96 views

rationality of Fano 3fold $X_{18}$

I need a reference for an explicit proof of the rationality of the Fano 3-fold $X_{18}$. By explicit I mean by a sequence of explicit birational transformations. Thank you!
5
votes
1answer
144 views

Automorphisms of Cartesian products

Let us consider the Cartesian product $X^r$, where $X$ is a smooth projective variety. There is a subgroup $Aut_{\Delta}(X^r)\subset Aut(X^r)$ of automorphisms of $X^r$ mapping a $k$-dimensional ...
1
vote
2answers
189 views

Automorphisms of locally trivial fibrations

Let $f:X\rightarrow Y$ be a locally trivial fibration with a variety $F$ as the fiber. Here $X, Y, F$ are smooth, projective varieties. Does any automorphism of $F$ induce an automorphism of $X$? In ...
11
votes
3answers
494 views

Automorphisms of cartesian products of curves

Let $C$ be a smooth projective curve. Is it true that $$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$ and in case, what would be a reference for this? Thanks.
80
votes
8answers
8k 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 ...
2
votes
1answer
159 views

Inverse image of a divisor

Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor. Let us look at the cycle $f^{-1}(D)\subset X$...
2
votes
2answers
141 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 $H^0(X,\...
2
votes
0answers
85 views

Resolution of indeterminacies for a map to Grassmannian of planes

Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
1
vote
1answer
141 views

Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$. Now, let $f:X\...
0
votes
0answers
57 views

Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...
4
votes
1answer
150 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
1answer
138 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 $X$, $\bar X\backslash X$...
0
votes
0answers
89 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 ...
2
votes
0answers
62 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, ...
5
votes
3answers
399 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
2answers
345 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
0answers
95 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 ...
6
votes
1answer
197 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 $\mathcal{I}_X/\mathcal{I}_X^...
14
votes
4answers
516 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 ...
1
vote
2answers
316 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 - 9\,...
3
votes
1answer
298 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 ...
4
votes
3answers
474 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 \mathbb{P}(1,...
3
votes
2answers
137 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 ...
4
votes
1answer
156 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 ...
5
votes
1answer
279 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 $q\in X$,...
4
votes
0answers
701 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 ...