All Questions
Tagged with cubics algebraic-geometry
35
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Are the 28 bitangents on quartic curves bounded distance away from each other?
We know that the 28 bitangents on a smooth plane quartic curve over $\mathbb{C}$ are all distinct. Are the bitangents bounded distance away from each other? More precisely, is there a constant $d>0$...
- 101
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26
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Question about real inflection points of cubic curves in P^2(C)
An elementary question about real inflection points of cubics:
Textbooks mention that non-singular cubics in $P^2(C)$ have 3 real and 6 complex inflections and show the Hesse normal form $ x^3 + y^3 +...
0
votes
1
answer
82
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What is a curve at $y=\infty$ mean?
On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown:
Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the '...
- 1,685
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50
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Why is the locus of points given two segments AB and CD such that APB=CPD give a degree-3 curve (in complex proj plane)?
The isoptic cubic is defined as the locus of points given two segments AB and CD (similarily oriented) such that APB=CPD (directed angles). By elementary geometry this would go through AB $\cap$ CD, ...
- 254
1
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37
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Proof of Thomson cubic pivotal property without coordinates
The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
- 254
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49
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For cubic surface, if $\dim(\operatorname{Sing}(X))\geq 1$ then a line is contained a in $\operatorname{Sing}(X)$
Let $X$ to be an irreducible cubic surface in $\mathbb{P}^3_{\mathbb{C}}$. Is it true that if $\dim(\operatorname{Sing}(X))\geq 1$, then a line is contained in $\operatorname{Sing}(X)$? I.e, does a ...
- 173
1
vote
1
answer
128
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Why are all real inflection points on a cubic projective algebraic curve on 1 line?
Say we have $C\subset \mathbb{CP}^2$, a smooth curve of degree 3. I am aware of the group structure on cubics, what I don't get, is why are all inflection points with only real coordinates lie on a ...
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39
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Disjoint exceptional lines on non-minimal cubic surface
A diagonal cubic surface $\sum_{i=0}^3a_iT_i^3=0$ is not minimal if, for example, $a_1a_2a_3^{-1}a_4^{-1}\in(k^*)^3$. This should be because there is an exceptional line $D$ such that no element in ...
- 115
-1
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1
answer
55
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Descartes folium
The geometry of Descartes' folium, $x^3+y^3=3axy$ has been well studied. Can someone tell me which geometric property characterizes the following cubic curve:
$$bx^3+y^3=3axy$$
The previous curve is a ...
- 1
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76
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For translation of axes, is there a definite equation for any of "the 27 lines" on the Clebsch Diagonal Cubic?
For the Clebsch Diagonal Cubic (related to a Quanta mag article on Hilbert's 13th Problem), I want to generate a point at will on any of these lines along the surface. Wolfram calls these "...
2
votes
1
answer
79
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Classification of curves passing through 7 points. (Hartshorne III ex 10.7)
This is the exercise III 10.7 in Hartshorne's Algebraic Geometry
I am not sure if I misunderstood the question.
The seven points of the projective plane over $\mathbb{F}_2$, I think, means
$\{[x_0,...
- 386
2
votes
2
answers
98
views
Given a cubic and a point S not on the cubic, how many tangent lines to F can we draw from S?
Let $F\in \mathbb{C}[x,y,z]$ be an irreducible homogeneous polynomial with total degree 3, defining a cubic in $\mathbb{CP}^2$. Given a point $S$ on the projective plane but not on $\mathbb{V}(F)$, ...
0
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1
answer
122
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Further information on the reduction of cubic equations to a system of two conic sections
This question follows on from one I have previously asked, How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam and I now would like some further advice on some ...
- 55
0
votes
1
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165
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For which points $P$ on a nonsingular cubic $C$ does there exist a nonsingular conic that intersects $C$ only at $P$?
For which points $P$ on a nonsingular cubic $C$ does there exist a nonsingular conic that intersects $C$ only at $P$?
By Bezout's theorem, we must have that $I(P,C \cap F)=6$, where $F$ is the ...
- 4,510
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General properties of cubic hypersurfaces
Is there any literature dealing with cubic hypersurfaces in full generality (over $\mathbb{C}$)? Couldn't find any.
We know everything about hyperplanes. We also know a lot of things about quadric ...
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