Plane quartic curve

In algebraic geometry, a plane quartic curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:

A

x

4

+

B

y

4

+

C

x

3

y

+

D

x

2

y

2

+

E

x

y

3

+

F

x

3

+

G

y

3

+

H

x

2

y

+

I

x

y

2

+

J

x

2

+

K

y

2

+

L

x

y

+

M

x

+

N

y

+

P

=

0

,

{\displaystyle Ax^{4}+By^{4}+Cx^{3}y+Dx^{2}y^{2}+Exy^{3}+Fx^{3}+Gy^{3}+Hx^{2}y+Ixy^{2}+Jx^{2}+Ky^{2}+Lxy+Mx+Ny+P=0,}

with at least one of A, B, C, D, E not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space ⁠

R

P

14

.

{\displaystyle \mathbb {RP} ^{14}.}

⁠ It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom.

A quartic curve can have a maximum of:

Four connected components

Twenty-eight bi-tangents

Three ordinary double points.

One may also consider quartic curves over other fields (or even rings), for instance the complex numbers. In this way, one gets Riemann surfaces, which are one-dimensional objects over ⁠

C

,

{\displaystyle \mathbb {C} ,}

⁠ but are two-dimensional over ⁠

R

.

{\displaystyle \mathbb {R} .}

⁠ An example is the Klein quartic.