Algebraic Geometry
Table of Contents
1. Line
1.1. Cartesian Coordinates
1.1.1. Standard form
- \(y=mx+n\)
1.1.2. General form
- \(ax+by+c=0\)
1.1.3. Intercept form
- \[ \frac{x}{a}+\frac{y}{b}=1 \]
1.1.4. Ratio form
- \[ \frac{y-y_0}{x-x_0}=k \]
1.1.5. Point-Gradient form
- \((y-y_0)=m(x-x_0)\)
1.1.6. Two-Point form
- \[ (y-y_1)=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \]
1.1.7. Point-Displacement Form
- \[ t=\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}=\dots \]
1.1.8. Parameterized Form
\[
\begin{align*} x&=at+x_0\\ y&=bt+y_0\\ z&=ct+z_0\\ &\dots \end{align*}\]
1.2. Vector
1.2.1. Vector Parameterized form
- \(\mathbf{x}=\mathbf{x}_0+k\mathbf{d}\)
- \(\mathbf{d}\) consists of the denominators from
((652778ff-870f-4b88-89de-08a76ccccd86)).
- \[ \mathbf{d}=\begin{bmatrix}a\\b\\c\\\vdots\end{bmatrix} \]
1.2.2. Affine Form
- Or Convex combination form, Barycentric form
- \(\mathbf{x}=t\mathbf{x_1}+(1-t)\mathbf{x_2}\)
1.2.3. Circle Secant Form
- Line that passes through two points on a circle, such that two
tangent lines at the points passes through some point \(\bf a\):
- \((\mathbf{a}-\mathbf{x_0})\cdot(\mathbf{x}-\mathbf{x_0})=r^2\)
- From the ((652781e2-c54f-43c7-8265-22d95d32c26c)) of the equation of circle.
- Although it started from a notion of tangent lines, but it also works for the points inside of a circle.
- Related to
1.2.4. Circle Tangent Form
- Tangent line of a circle
- \((\mathbf{x}_1-\mathbf{x}_0)\cdot(\mathbf{x}-\mathbf{x_0})=r^2\) where \(\mathbf{x}_1\) is a point on the circle.
1.3. Plücker Coordinates
- Six homogeneous coordinates that corresponds to a line in projective 3-space \(\mathbb{P}^3\).
- It consists of a displacement \(\mathbf{d}\) and a moment
\(\mathbf{m}\): \((d_1: d_2: d_3: m_1: m_2: m_3)\).
- For two points \(\mathbf{x}, \mathbf{y}\) on the line,
\(\mathbf{d} = \mathbf{y} - \mathbf{x}\) and
\(\mathbf{m} = \mathbf{x}\times \mathbf{y}\).
- The moment is perpendicular to the plane that passes through the line and the origin.
Alternatively, since the equation of line can be described as the intersection of two planes: \[
\begin{cases} \mathbf{a\cdot x} + a = 0\\ \mathbf{b\cdot x} + b = 0 \end{cases}\]
- \(\mathbf{d} = \mathbf{a\times b}\) and \(\mathbf{m} = a\mathbf{b} - b\mathbf{a}\).
- For two points \(\mathbf{x}, \mathbf{y}\) on the line,
\(\mathbf{d} = \mathbf{y} - \mathbf{x}\) and
\(\mathbf{m} = \mathbf{x}\times \mathbf{y}\).
1.4. Singularity
1.4.1. Tacnode
- Point of Osculation, Double Cusp
1.4.1.1. Definition
Singular point of a curve, at which osculating circles at that point are tangent.
1.4.1.2. Example
\[ y^2 - x^4 = 0 \] is the canonical example
1.4.2. Acnode
- Isolated Point, Hermit Point
1.4.2.1. Definition
Isolated point in the solution set of a polynomial equation in two variables.
1.4.2.2. Example
\[ f(x, y) = y^2 + x^2 - x^3 = 0 \] attains an acnode at the origin.
1.4.3. Crunode
- Node, Ordinary Double Point
1.4.3.1. Definition
A point where a curve intersects itself, with both branches have distinct tangent lines.
1.4.3.2. Properties
Both partial derivatives with respect to \(x\) and \(y\) vanishes, and the Hessian matrix will have both positive and negative eigenvalues.
1.4.3.3. Example
\[y^2 - x^2(x+1) = 0\] attains crunode at the origin.
2. Fermat Curve
Algebraic curve in the complex projective plane, defined in homogeneous coordinates \( (X\mathpunct{:}Y\mathpunct{:}Z) \) by the Fermat equation \( X^n + Y^n = Z^n \).
3. Plane
3.1. Orthogonal Form
- \(\mathbf{n}\cdot(\mathbf{x}-\mathbf{x}_0)=0\)
4. Algebraic Variety
- Classically, a set of solutions of a system of polynomial equations.
4.1. Algebraic Set
- The zero-locus in affine \(n\)-space \(\mathbb{A}^n\) of the set \(S\) of polynomials in \(K[x_1, \dots, x_n]\): \[ V(S) := \{x\in \mathbb{A}^n \mid \forall f\in S, f(x) = 0\}. \]
4.1.1. Irreducibility
- A nonempty affine algebraic set \(V(S)\) is called irreducible if it cannot be written as the union of two proper algebraic subsets.
4.2. Defintion
- Set of roots of the ideals, in an algebraically closed field.
4.2.1. Affine Variety
- Irreducible affine algebraic set \(V(S)\) for any set of polynomials \(S\)
4.2.2. Abstract Definition
- integral, separated scheme of finite type over an algebraically closed field
5. Bezout's Theorem
The number of common zeroes of n polynomials in n indeterminates, with the domain of complex projective space, is equal to the product of the degrees of the polynomials accounting for the multiplicity.
6. Hilbert's Nullstellensatz
6.1. Statement
For a field \(k\) and its field extension \(K\), the ideal \(I\) of the polynomial ring \(k[X_1, \dots, X_n]\) and the algebraic set \(\mathrm{V}(I)\subset K^n\), if \(p\) vanished on \(\mathrm{V}(I)\), then \(\exists r\in \mathbb{N}\) such that \(p^r \in I\).
Equivalently, \[ \mathrm{I}(\mathrm{V}(J)) = \sqrt{J} \] where \(\sqrt{J}\) denotes the radical of \(J\), and \(\mathrm{I}(U)\) is the ideal of all polynomials that vanish on the set \(U\).
7. Riemann's Existence Theorem
7.1. Statement
Every compact Riemann surface is an algebraic curve. There exists smooth embedding of all compact Riemann surfaces into the complex projective space, such that the embedding is isomorphic to a zero set of polynomial equations.
7.2. Examples
For any function \(f: \mathbb{CP}^1 \to \mathbb{CP}^1\) meromorphic on a Riemann sphere
- The graph \(\{(x,y)\in \mathbb{CP}^2 \mid y = f(x)\}\) is guaranteed to be a compact Riemann surface, since \(f(x)\) is always rational.
\[f: \mathbb{C}\backslash \Lambda\to \mathbb{CP}^2\] such that \[ f: z\mapsto (\weierp(z), \weierp'(z)) \] where \(\weierp\) is the Weierstrass elliptic funciton with standard lattice. The graph of the range is a complex torus given by \[\{(x,y)\in \mathbb{CP}^2 \mid y^2 = 4x^3 + 4x\}\] that is, a zero set of polynomial equations.
7.3. Implications
It is the link between complex analysis and algebraic geometry.