Исаак Ньютон получил две классификации кубик [1] [2] [2] аффинная классификация кубик[3]
Теорема. Существуют 59 семейств аффинных классов эквивалентности неприводимых кубик :
15 классов модальности 0;
23 семейства (классов) модальности 1;
16 семейств модальности 2;
5 семейств модальности 3;
эти семейства представлены в следующем списке канонических уравнений.
Порядок перечисления семейств аффинных классов принадлежит Ньютону, для удобства он сохранён в этом списке. В каждом пункте списка указана размерность множества кубик, принадлежащих этому семейству аффинных классов. Например, каждая кубика аффинного класса с номером 1.1 аффинно эквивалентна кубике
y
2
=
x
3
{\displaystyle y^{2}=x^{3}}
R
P
9
{\displaystyle \mathbb {R} P^{9}}
dim
=
5
{\displaystyle \dim =5}
(
x
−
1
)
y
2
−
a
x
2
y
+
x
3
=
0
{\displaystyle (x-1)y^{2}-ax^{2}y+x^{3}=0}
0
<
a
<
2
{\displaystyle 0<a<2}
R
P
9
{\displaystyle \mathbb {R} P^{9}}
dim
=
7
{\displaystyle \dim =7}
Рис. 1: Классы кубик с точкой возврата Классы, полученные из кубики с точкой возврата, см. рис. 1.
1.1.
y
2
=
x
3
{\displaystyle y^{2}=x^{3}}
dim
=
5
{\displaystyle \dim =5}
1.2.
y
=
x
3
{\displaystyle y=x^{3}}
dim
=
5
{\displaystyle \dim =5}
1.3.
x
2
y
=
1
{\displaystyle x^{2}y=1}
dim
=
6
{\displaystyle \dim =6}
1.4.
x
2
y
=
1
−
x
{\displaystyle x^{2}y=1-x}
dim
=
6
{\displaystyle \dim =6}
1.5.
(
x
−
1
)
y
2
+
x
3
=
0
{\displaystyle (x-1)y^{2}+x^{3}=0}
dim
=
6
{\displaystyle \dim =6}
1.6.
(
x
+
1
)
y
2
=
x
3
{\displaystyle (x+1)y^{2}=x^{3}}
dim
=
6
{\displaystyle \dim =6}
1.7.
(
x
−
1
)
y
2
−
a
x
2
y
+
x
3
=
0
{\displaystyle (x-1)y^{2}-ax^{2}y+x^{3}=0}
0
<
a
<
2
{\displaystyle 0<a<2}
dim
=
7
{\displaystyle \dim =7}
1.8.
y
2
−
x
2
y
−
x
3
=
0
{\displaystyle y^{2}-x^{2}y-x^{3}=0}
dim
=
6
{\displaystyle \dim =6}
1.9.
(
x
+
1
)
y
2
−
a
x
2
y
+
x
3
=
0
{\displaystyle (x+1)y^{2}-ax^{2}y+x^{3}=0}
a
>
0
{\displaystyle a>0}
dim
=
7
{\displaystyle \dim =7}
Рис. 2: Классы кубик с петлёй Классы, полученные из кубики с петлёй, см. рис. 2.
2.1.
y
2
=
x
2
(
x
+
1
)
{\displaystyle y^{2}=x^{2}(x+1)}
dim
=
6
{\displaystyle \dim =6}
2.2.
(
1
−
x
a
)
y
2
=
x
2
(
x
+
1
)
{\displaystyle \left(1-{\frac {x}{a}}\right)y^{2}=x^{2}\left(x+1\right)}
a
>
0
{\displaystyle a>0}
dim
=
7
{\displaystyle \dim =7}
2.3.
(
1
−
x
)
y
2
=
x
2
{\displaystyle \left(1-x\right)y^{2}=x^{2}}
dim
=
6
{\displaystyle \dim =6}
2.4.
(
1
−
x
a
)
y
2
=
x
2
(
1
−
x
)
{\displaystyle \left(1-{\frac {x}{a}}\right)y^{2}=x^{2}\left(1-x\right)}
0
<
a
<
1
{\displaystyle 0<a<1}
dim
=
6
{\displaystyle \dim =6}
2.5.
(
x
+
1
)
(
x
−
1
)
y
+
1
=
0
{\displaystyle (x+1)(x-1)y+1=0}
dim
=
6
{\displaystyle \dim =6}
2.6.
(
x
a
+
1
)
y
2
=
x
2
(
x
+
1
)
{\displaystyle \left({\frac {x}{a}}+1\right)y^{2}=x^{2}\left(x+1\right)}
a
>
1
{\displaystyle a>1}
dim
=
7
{\displaystyle \dim =7}
2.7.
(
x
−
1
)
y
2
=
b
x
(
x
+
a
)
(
y
−
x
)
{\displaystyle \left(x-1\right)y^{2}=bx(x+a)(y-x)}
a
>
0
{\displaystyle a>0}
0
≤
b
<
4
{\displaystyle 0\leq b<4}
dim
=
8
{\displaystyle \dim =8}
2.8.
(
x
−
1
)
y
2
=
a
x
(
y
−
x
)
{\displaystyle (x-1)y^{2}=ax(y-x)}
a
>
0
{\displaystyle a>0}
dim
=
7
{\displaystyle \dim =7}
2.9.
x
y
=
(
x
−
1
)
3
{\displaystyle xy=(x-1)^{3}}
dim
=
6
{\displaystyle \dim =6}
2.10.
(
x
−
1
)
y
2
=
a
x
(
y
−
x
)
{\displaystyle (x-1)y^{2}=ax(y-x)}
a
<
−
4
{\displaystyle a<-4}
dim
=
7
{\displaystyle \dim =7}
2.11.
(
1
−
x
)
y
2
=
b
x
(
x
−
a
)
(
y
−
x
)
{\displaystyle \left(1-x\right)y^{2}=bx\left(x-a\right)\left(y-x\right)}
a
>
1
{\displaystyle a>1}
b
>
0
{\displaystyle b>0}
dim
=
8
{\displaystyle \dim =8}
2.12.
(
x
+
1
)
(
x
−
a
)
y
+
x
=
0
{\displaystyle (x+1)(x-a)y+x=0}
a
>
0
{\displaystyle a>0}
dim
=
7
{\displaystyle \dim =7}
2.13.
(
x
+
1
)
(
x
−
a
)
y
+
x
2
=
0
{\displaystyle (x+1)(x-a)y+x^{2}=0}
a
>
0
{\displaystyle a>0}
a
≠
1
{\displaystyle a\neq 1}
dim
=
7
{\displaystyle \dim =7}
2.14.
(
x
a
+
1
)
y
2
−
b
x
2
y
=
x
2
(
x
+
1
)
{\displaystyle \left({\frac {x}{a}}+1\right)y^{2}-bx^{2}y=x^{2}\left(x+1\right)}
a
>
0
{\displaystyle a>0}
b
>
0
{\displaystyle b>0}
dim
=
8
{\displaystyle \dim =8}
Рис. 3: Классы кубик с изолированной точкой Классы, полученные из кубики с изолированной точкой, см. рис. 3, где кубики семейств с номерами 3.1, 3.2, 3.4 - 3.8, 3.10 - 3.12 имеют изолированную точку в начале координат
O
=
(
0
,
0
)
{\displaystyle O=(0,0)}
x
=
0
{\displaystyle x=0}
z
=
0
{\displaystyle z=0}
(
0
:
1
:
0
)
{\displaystyle (0:1:0)}
3.1.
y
2
=
x
2
(
x
−
1
)
{\displaystyle y^{2}=x^{2}(x-1)}
dim
=
6
{\displaystyle \dim =6}
3.2.
(
1
−
x
a
)
y
2
=
x
2
(
x
−
1
)
{\displaystyle \left(1-{\frac {x}{a}}\right)y^{2}=x^{2}\left(x-1\right)}
a
>
1
{\displaystyle a>1}
dim
=
7
{\displaystyle \dim =7}
3.3.
(
x
2
+
1
)
y
=
x
2
{\displaystyle (x^{2}+1)y=x^{2}}
dim
=
6
{\displaystyle \dim =6}
3.4.
(
x
a
+
1
)
y
2
=
−
x
2
(
x
+
1
)
{\displaystyle \left({\frac {x}{a}}+1\right)y^{2}=-x^{2}\left(x+1\right)}
0
<
a
<
1
{\displaystyle 0<a<1}
dim
=
7
{\displaystyle \dim =7}
3.5.
(
x
+
1
)
y
2
=
−
x
2
{\displaystyle \left(x+1\right)y^{2}=-x^{2}}
dim
=
6
{\displaystyle \dim =6}
3.6.
(
x
a
+
1
)
y
2
=
x
2
(
x
−
1
)
{\displaystyle \left({\frac {x}{a}}+1\right)y^{2}=x^{2}\left(x-1\right)}
0
<
a
<
1
3
{\displaystyle 0<a<{\frac {1}{3}}}
dim
=
7
{\displaystyle \dim =7}
3.7.
(
3
x
+
1
)
y
2
=
x
2
(
x
−
1
)
{\displaystyle \left(3x+1\right)y^{2}=x^{2}\left(x-1\right)}
dim
=
6
{\displaystyle \dim =6}
3.8.
(
x
a
+
1
)
y
2
=
x
2
(
x
−
1
)
{\displaystyle \left({\frac {x}{a}}+1\right)y^{2}=x^{2}\left(x-1\right)}
a
>
1
3
{\displaystyle a>{\frac {1}{3}}}
dim
=
7
{\displaystyle \dim =7}
3.9.
y
(
x
2
+
a
x
+
1
)
=
x
{\displaystyle y(x^{2}+ax+1)=x}
0
≤
a
<
2
{\displaystyle 0\leq a<2}
dim
=
7
{\displaystyle \dim =7}
3.10.
(
1
−
x
)
y
2
+
b
x
(
x
−
a
)
(
y
−
x
)
=
0
{\displaystyle (1-x)y^{2}+bx(x-a)(y-x)=0}
0
<
a
<
1
{\displaystyle 0<a<1}
0
<
b
<
4
{\displaystyle 0<b<4}
dim
=
8
{\displaystyle \dim =8}
3.11.
(
x
+
1
)
y
2
+
a
x
(
y
+
x
)
=
0
{\displaystyle (x+1)y^{2}+ax(y+x)=0}
0
<
a
<
4
{\displaystyle 0<a<4}
dim
=
7
{\displaystyle \dim =7}
3.12.
(
x
+
1
)
y
2
−
b
x
(
x
−
a
)
(
y
+
x
)
=
0
{\displaystyle \left(x+1\right)y^{2}-bx(x-a)(y+x)=0}
a
>
0
{\displaystyle a>0}
b
>
0
{\displaystyle b>0}
a
<
(
a
+
1
)
(
b
+
2
−
b
2
+
4
b
)
2
(
b
+
4
)
−
(
a
+
1
)
(
b
+
2
+
b
2
+
4
b
)
{\displaystyle a<{\frac {(a+1)(b+2-{\sqrt {b^{2}+4b}})}{2(b+4)-(a+1)\left(b+2+{\sqrt {b^{2}+4b}}\right)}}}
dim
=
8
{\displaystyle \dim =8}
Рис. 4: Классы простой кубики Классы, полученные из простой кубики, см. рис. 4.
4.1.
y
2
=
x
[
(
x
−
a
)
2
+
1
]
{\displaystyle y^{2}=x\left[\left(x-a\right)^{2}+1\right]}
a
∈
R
{\displaystyle a\in \mathbb {R} }
dim
=
7
{\displaystyle \dim =7}
4.2.
x
y
2
=
−
(
x
+
a
)
[
(
x
−
b
)
2
+
1
]
{\displaystyle xy^{2}=-(x+a)\left[\left(x-b\right)^{2}+1\right]}
a
>
0
{\displaystyle a>0}
b
∈
R
{\displaystyle b\in \mathbb {R} }
dim
=
8
{\displaystyle \dim =8}
4.3.
x
y
2
=
−
[
(
x
−
a
)
2
+
1
]
{\displaystyle xy^{2}=-\left[\left(x-a\right)^{2}+1\right]}
a
∈
R
{\displaystyle a\in \mathbb {R} }
dim
=
7
{\displaystyle \dim =7}
4.4.
x
y
2
=
−
(
x
−
a
)
[
(
x
−
b
)
2
+
1
]
{\displaystyle xy^{2}=-(x-a)\left[\left(x-b\right)^{2}+1\right]}
a
>
0
{\displaystyle a>0}
b
<
a
2
−
4
4
a
{\displaystyle b<{\frac {a^{2}-4}{4a}}}
dim
=
8
{\displaystyle \dim =8}
4.5.
x
y
2
=
−
(
x
−
a
)
[
(
x
−
a
2
−
4
4
a
)
2
+
1
]
{\displaystyle xy^{2}=-(x-a)\left[\left(x-{\frac {a^{2}-4}{4a}}\right)^{2}+1\right]}
a
>
0
{\displaystyle a>0}
dim
=
7
{\displaystyle \dim =7}
4.6.
x
y
2
=
−
(
x
−
a
)
[
(
x
−
b
)
2
+
1
]
{\displaystyle xy^{2}=-(x-a)\left[\left(x-b\right)^{2}+1\right]}
a
>
0
{\displaystyle a>0}
b
>
a
2
−
4
4
a
{\displaystyle b>{\frac {a^{2}-4}{4a}}}
dim
=
8
{\displaystyle \dim =8}
4.7.
x
y
2
=
c
[
(
x
−
a
)
2
+
1
]
(
y
+
x
−
b
)
{\displaystyle xy^{2}=c[(x-a)^{2}+1](y+x-b)}
a
∈
R
{\displaystyle a\in \mathbb {R} }
b
≥
0
{\displaystyle b\geq 0}
−
4
<
c
<
0
{\displaystyle -4<c<0}
dim
=
9
{\displaystyle \dim =9}
4.8.
x
y
2
=
−
4
[
(
x
−
a
)
2
+
1
]
(
y
+
x
−
b
)
{\displaystyle xy^{2}=-4[(x-a)^{2}+1](y+x-b)}
a
∈
R
{\displaystyle a\in \mathbb {R} }
b
≥
0
{\displaystyle b\geq 0}
b
≠
2
a
{\displaystyle b\neq 2a}
dim
=
8
{\displaystyle \dim =8}
4.9.
x
y
2
=
c
[
(
x
−
a
)
2
+
1
]
(
y
+
x
−
b
)
{\displaystyle xy^{2}=c[(x-a)^{2}+1](y+x-b)}
a
∈
R
{\displaystyle a\in \mathbb {R} }
b
≥
0
{\displaystyle b\geq 0}
c
(
c
+
4
)
(
a
2
+
1
)
<
a
c
−
2
b
{\displaystyle {\sqrt {c(c+4)(a^{2}+1)}}<ac-2b}
0
<
c
<
2
[
a
2
−
a
b
+
1
+
(
a
2
−
a
b
+
1
)
2
+
b
2
]
{\displaystyle 0<c<2\left[a^{2}-ab+1+{\sqrt {(a^{2}-ab+1)^{2}+b^{2}}}\right]}
a
c
+
2
b
+
(
a
c
+
2
b
)
2
−
c
(
c
+
4
)
(
a
2
+
1
)
c
+
4
<
(
c
+
4
)
(
a
2
+
1
)
(
β
−
b
)
(
2
a
−
b
)
2
−
(
c
+
4
)
(
a
2
+
1
)
(
α
+
1
)
{\displaystyle {\frac {ac+2b+{\sqrt {(ac+2b)^{2}-c(c+4)(a^{2}+1)}}}{c+4}}<{\frac {(c+4)(a^{2}+1)(\beta -b)}{(2a-b)^{2}-(c+4)(a^{2}+1)(\alpha +1)}}}
c
(
c
+
4
)
(
a
2
+
1
)
<
a
c
+
2
b
{\displaystyle {\sqrt {c(c+4)(a^{2}+1)}}<ac+2b}
α
=
c
−
c
2
+
4
c
2
{\displaystyle \alpha ={\frac {c-{\sqrt {c^{2}+4c}}}{2}}}
β
=
−
c
[
a
−
a
(
c
+
2
)
+
b
c
2
+
4
c
]
{\displaystyle \beta =-c\left[a-{\frac {a(c+2)+b}{\sqrt {c^{2}+4c}}}\right]}
dim
=
9
{\displaystyle \dim =9}
Рис. 5: Классы кубики с овалом Классы, полученные из кубики с овалом, см. рис. 5.
5.1.
y
2
=
x
(
x
+
1
)
(
x
+
a
)
{\displaystyle y^{2}=x(x+1)(x+a)}
a
>
1
{\displaystyle a>1}
dim
=
7
{\displaystyle \dim =7}
5.2.
x
y
2
=
−
(
x
+
1
)
(
x
+
a
)
(
x
+
b
)
{\displaystyle xy^{2}=-(x+1)(x+a)(x+b)}
1
<
a
<
b
{\displaystyle 1<a<b}
dim
=
8
{\displaystyle \dim =8}
5.3.
x
y
2
=
−
(
x
+
1
)
(
x
+
a
)
{\displaystyle xy^{2}=-(x+1)(x+a)}
a
>
1
{\displaystyle a>1}
dim
=
7
{\displaystyle \dim =7}
5.4.
x
y
2
=
−
(
x
+
a
)
(
x
+
1
)
(
x
−
b
)
{\displaystyle xy^{2}=-(x+a)(x+1)(x-b)}
a
>
1
{\displaystyle a>1}
b
>
0
{\displaystyle b>0}
dim
=
8
{\displaystyle \dim =8}
5.5.
x
y
2
=
(
x
+
1
)
(
x
−
a
)
{\displaystyle xy^{2}=(x+1)(x-a)}
a
>
0
{\displaystyle a>0}
dim
=
7
{\displaystyle \dim =7}
5.6.
x
y
2
=
−
(
x
+
1
)
(
x
−
a
)
(
x
−
b
)
{\displaystyle xy^{2}=-(x+1)(x-a)(x-b)}
0
<
a
<
b
{\displaystyle 0<a<b}
dim
=
8
{\displaystyle \dim =8}
5.7.
x
y
2
=
−
(
x
−
1
)
(
x
−
a
)
{\displaystyle xy^{2}=-(x-1)(x-a)}
a
>
1
{\displaystyle a>1}
dim
=
7
{\displaystyle \dim =7}
5.8.
x
y
2
=
(
x
−
a
)
(
x
−
1
)
(
x
−
b
)
{\displaystyle xy^{2}=(x-a)(x-1)(x-b)}
0
<
a
<
1
<
b
{\displaystyle 0<a<1<b}
b
>
(
a
+
1
)
2
{\displaystyle b>({\sqrt {a}}+1)^{2}}
dim
=
8
{\displaystyle \dim =8}
5.9.
x
y
2
=
(
x
−
a
)
(
x
−
1
)
[
x
−
(
a
+
1
)
2
]
{\displaystyle xy^{2}=(x-a)(x-1)[x-({\sqrt {a}}+1)^{2}]}
0
<
a
<
1
{\displaystyle 0<a<1}
dim
=
7
{\displaystyle \dim =7}
5.10.
x
y
2
=
(
x
−
a
)
(
x
−
1
)
(
x
−
b
)
{\displaystyle xy^{2}=(x-a)(x-1)(x-b)}
0
<
a
<
1
<
b
{\displaystyle 0<a<1<b}
b
<
(
a
+
1
)
2
{\displaystyle b<({\sqrt {a}}+1)^{2}}
dim
=
8
{\displaystyle \dim =8}
5.11.
x
y
2
=
c
(
x
+
1
)
(
x
+
a
)
(
x
+
y
−
b
)
{\displaystyle xy^{2}=c(x+1)(x+a)(x+y-b)}
a
>
1
{\displaystyle a>1}
b
>
−
1
{\displaystyle b>-1}
−
4
<
c
<
0
{\displaystyle -4<c<0}
dim
=
9
{\displaystyle \dim =9}
5.12.
x
y
2
=
b
(
x
+
1
)
(
x
+
y
−
a
)
{\displaystyle xy^{2}=b(x+1)(x+y-a)}
a
>
−
1
{\displaystyle a>-1}
b
∈
R
{\displaystyle b\in \mathbb {R} }
dim
=
8
{\displaystyle \dim =8}
5.13.
x
y
2
=
c
(
x
+
1
)
(
x
−
a
)
(
x
+
y
−
b
)
{\displaystyle xy^{2}=c(x+1)(x-a)(x+y-b)}
a
>
0
{\displaystyle a>0}
−
1
<
b
<
a
{\displaystyle -1<b<a}
c
>
0
{\displaystyle c>0}
dim
=
9
{\displaystyle \dim =9}
5.14.
x
y
2
=
−
4
(
x
+
1
)
(
x
+
a
)
(
x
+
y
−
b
)
{\displaystyle xy^{2}=-4(x+1)(x+a)(x+y-b)}
a
>
1
{\displaystyle a>1}
b
>
−
1
{\displaystyle b>-1}
dim
=
8
{\displaystyle \dim =8}
5.15.
x
y
2
=
c
(
x
−
a
)
(
x
−
1
)
(
x
+
y
−
b
)
{\displaystyle xy^{2}=c(x-a)(x-1)(x+y-b)}
0
<
a
<
1
<
b
{\displaystyle 0<a<1<b}
c
>
0
{\displaystyle c>0}
a
c
(
β
−
b
)
β
2
−
c
[
a
(
α
+
1
)
−
(
a
+
1
)
(
β
−
b
)
]
>
c
(
a
+
1
)
+
4
b
+
D
2
(
c
+
4
)
{\displaystyle {\frac {ac\left(\beta -b\right)}{\beta ^{2}-c\left[a\left(\alpha +1\right)-(a+1)\left(\beta -b\right)\right]}}>{\frac {c(a+1)+4b+{\sqrt {\mathcal {D}}}}{2(c+4)}}}
D
=
[
c
(
a
−
1
)
+
4
b
]
2
+
16
c
(
b
−
a
)
{\displaystyle {\mathcal {D}}=[c(a-1)+4b]^{2}+16c(b-a)}
α
=
c
−
c
2
+
4
c
2
{\displaystyle \alpha ={\frac {c-{\sqrt {c^{2}+4c}}}{2}}}
β
=
c
[
b
+
(
a
+
1
)
(
α
+
1
)
]
c
−
2
α
{\displaystyle \beta ={\frac {c\left[b+(a+1)\left(\alpha +1\right)\right]}{c-2\alpha }}}
dim
=
9
{\displaystyle \dim =9}
↑
Newton I. "Enumeratio linearum tertii ordinis". — in "The mathematical papers of Isaac Newton" (D.T. Whiteside, ed.): Cambridge Univ. Press , V. 7, 1976, pp. 565-645.Д. Д. Мордухай-Болтовского ), 1937, стр. 194—209, доступно постранично on-line на Архивированная копия (неопр.) . Дата обращения: 8 февраля 2016. Архивировано 12 июня 2008 года.
↑ 1 2
Newton I. "The final 'Geometriæ libri duo' ". — in "The mathematical papers of Isaac Newton" (D.T. Whiteside, ed.): Cambridge Univ. Press , V. 7, 1976, pp. 402-469.
↑ Корчагин А. Б., Ньютонова и аффинная классификации нераспадающихся кубик, Алгебра и анализ, Т. 24(2012), № 5, стр. 94–123. Engl. transl.: Korchagin A. B., Newtonian and affine classifications of irreducible cubics, St. Petersburg Math. J., Vol. 24, 2013, pp. 759-781.
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