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Results tagged with complex-geometry
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user 31356
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
10
votes
1
answer
411
views
Kelly's theorem about a resolution of the Sylvester–Gallai problem
Kelly's theorem states:
Every finite point set of complex space such that the line joining any two points from this set contains at least one more point from this set (every Sylvester-Gallai configara …
- 2,576
3
votes
1
answer
240
views
Linear homogenous polynomials that generates several quadratic polynomials
This is a generalization of this question.
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f_1, \ldots, f_s$ be a homogenous quadrati …
- 2,576
2
votes
0
answers
205
views
Sylvester-Gallai-type theorem for quadratic polynomials
Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ fr …
- 2,576
2
votes
3
answers
331
views
Linear homogenous polynomials that generates one quadratic polynomial
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and f …
- 2,576
2
votes
Accepted
Kelly's theorem about a resolution of the Sylvester–Gallai problem
I have found: https://arxiv.org/abs/1211.0330
Also https://arxiv.org/pdf/math/0403023.pdf (thank to Mike Miller).
- 2,576
0
votes
0
answers
105
views
Kelly's theorem for quadratic polynomials
Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$.
Assume that these polynomials are pairwise coprime.
Denote $P:= f_1 \cdot f_2 \ldots \cdo …
- 2,576
0
votes
0
answers
288
views
Union of varieties
Let $Q_1, \ldots, Q_k$ and $P_1,\ldots, P_m$ be irredicable homogenous polynomials in $\mathbb{C}[x_0,\ldots, x_n]$ such that $V(Q_1, \ldots, Q_k) \subseteq \cup_i V(P_i)$. Here $V$ is projective vari …
- 2,576
0
votes
Accepted
Linear homogenous polynomials that generates one quadratic polynomial
We will assume that $f$ is irreducible (if $f$ is not irreducible then in fact the argument of Zach Teitler's answer works).
Consider $M:= f \cap P_1$ (I mean the intersection of the zeros $f$ and $P …
- 2,576
0
votes
Accepted
Linear homogenous polynomials that generates several quadratic polynomials
Yes.
1) A quadratic homogenous polynomial $f$ (over $\mathbb{C}$) is irreducible iff $\text{rk}(f) \ge 3$. Here $\text{rk}(f)$ is the rank of $f$ as a quadratic form. Indeed, if $\text{rk}(f) < 3$ …
- 2,576