Question

Let \(X=C=\mathbb{P}^{1}, k(X)=k(t)\), where \(t=X_{1} / X_{2}, X_{1}, X_{2}\) homogeneous coordinates on \(\mathbb{P}^{1}\). (a) Calculate \(\operatorname{div}(t)\). (b) Calculate \(\operatorname{div}(f / g), f, g\) relatively prime in \(k[t]\). (c) Prove Proposition 1 directly in this case.

Short answer

#Answer# (a) The divisor of the coordinate function \(t\) is \(\operatorname{div}(t) = (0:1) - (1:0)\). (b) The divisor of the quotient of relatively prime polynomials \(f/g\) is \(\operatorname{div}(f/g) = \operatorname{div}(f) - \operatorname{div}(g)\). (c) Proposition 1 is proven directly for this case as we showed that the degree of the divisor of a rational function on \(\mathbb{P}^1\) is always zero. Specifically, \(\operatorname{deg}(\operatorname{div}(f/g)) = \operatorname{deg}(f) - \operatorname{deg}(g) = 0\).

Step-by-step solution

Find the zeros and poles of \(t\).

First, we need to find the zeros and poles of the function \(t\). To find the zeros, we need to look for points where \(X_1=0\). This occurs when \(t = 0\), which represents the point \((0:1)\) in projective coordinates. To find the poles, we need to find points where \(X_2 = 0\). This happens when \(t = \infty\), which represents the point \((1:0)\) in projective coordinates.

Determine the degree of zeros and poles of \(t\).

Since \(t\) has degree one as a polynomial, the divisor of \(t\) is \(\operatorname{div}(t) = (0:1) - (1:0)\).

Find the zeros and poles of \(f\) and \(g\).

To find the zeros and poles of the polynomials \(f\) and \(g\), we need to factor them into their irreducible factors and find the projective coordinates of their zeros and poles. As \(f\) and \(g\) are relatively prime, their zeros and poles are disjoint. Therefore, we can write \(f = \prod{P_i}\) and \(g = \prod{Q_j}\), where \(P_i\) and \(Q_j\) are the irreducible factors of \(f\) and \(g\), respectively. The zeros and poles of \(f\) are given by the projective coordinates of the zeros of its irreducible factors \(P_i\), while the zeros and poles of \(g\) are given by the projective coordinates of the zeros of its irreducible factors \(Q_j\).

Determine the divisor of \(f/g\).

Since we have found the zeros and poles of \(f\) and \(g\), we can determine the divisor of \(f/g\). The divisor of \(f/g\) is given by the difference of the divisors of \(f\) and \(g\): \(\operatorname{div}(f/g) = \operatorname{div}(f) - \operatorname{div}(g)\).

Proving Proposition 1 directly in this case.

Proposition 1 states that the degree of the divisor of a rational function is zero. We need to show this for the function \(f/g\) in this case. Since \(\operatorname{div}(f/g)\) is formed from the difference of the divisors of \(f\) and \(g\), the degree of \(\operatorname{div}(f/g)\) is equal to the sum of the degrees of the zeros and poles of \(f\) minus the sum of the degrees of the zeros and poles of \(g\). This is equal to the degree of \(f\) minus the degree of \(g\). Since \(f\) and \(g\) are relatively prime, their degrees are integers. Therefore, \(\operatorname{deg}(\operatorname{div}(f/g)) = \operatorname{deg}(f) - \operatorname{deg}(g) = 0\). This proves Proposition 1 directly for this case.

Key concepts

Divisors

In algebraic geometry, divisors play a crucial role in understanding the properties of a variety. Specifically, they provide a way to track the zeros and poles of rational functions across the space. A divisor on a projective space \mathbb{P}^n\ is an algebraic tool that helps mathematicians keep track of where a function is zero and where it "blows up" to infinity.
To illustrate, when working with rational functions like \(f/g\) on a projective line, their divisors \( \operatorname{div}(f/g) \) are computed as the difference between the zeros (points where the function is zero) and the poles (points where the function is undefined due to division by zero). For functions on projective coordinates such as \( t = X_1 / X_2 \), a divisor succinctly captures this balance, providing a way to quantify these points algebraically.

• A zero is a point where the function equals zero.
• A pole is a point where the function becomes infinite.

Understanding divisors is fundamental to analyzing the algebraic structures on varieties, providing deep insights into the geometry of the underlying space.

Projective Space

Projective space, denoted as \mathbb{P}^{n}\, is one of the most powerful concepts in algebraic geometry. It extends the concept of usual affine space by adding points at infinity, which allows mathematicians to tackle problems in a more comprehensive and symmetric way. This space is especially useful for studying properties that remain invariant under various transformations.
For example, \mathbb{P}^1\, also known as the projective line, is essentially a line with the addition of a single point at infinity, often denoted by \(t = \infty\). Here, the points are represented in homogeneous coordinates as \( (X_1 : X_2) \), which provide a natural way to handle intersections and parallels in geometry.

• In projective space, every line intersects at a point, even parallel ones, due to the addition of points at infinity.
• This helps maintain the continuity and uniformity of geometric transformations.

Overall, projective space creates a framework where each point and line can be treated evenly, thereby simplifying many complex algebraic geometry problems.

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in algebra that refers to the highest power of the variable present in the polynomial expression. In algebraic geometry, the degree plays an important role in defining the characteristics and behaviors of algebraic varieties.
For instance, when discussing the function \(t = X_1 / X_2 \) within the context of projective space, \mathbb{P}^1\, the degree indicates the zero and pole counts for the function. Here, the polynomial's degree helps determine the order in which zeros and poles appear and provides insight into their balancing act in a specific algebraic divisor.

• The degree of \(t\) as a polynomial is 1, indicating a linear change in behavior across the projective line.
• In the realm of rational functions, the degree also contributes to understanding divisors and their behavior.

Thus, the degree of a polynomial sheds light on how algebraic functions behave and interact with the structures in projective space.

Rational Functions

Rational functions are quotients of two polynomials and are extensively used in algebraic geometry to express functions on varieties. They provide a natural way to generalize and extend the notion of polynomials, allowing the study of more intricate relationships and properties of algebraic shapes.
For a function expressed as \( f/g \, the characteristics of this function are deeply tied to the properties of its numerator \( f \) and denominator \( g \). These functions may exhibit zeros (where the function evaluates to zero) and poles (where the function becomes undefined), both of which are captured via divisors.

• The set of rational functions provides a rich structure enabling the study of various algebraic and geometric properties in projective spaces.
• The degrees of the polynomials \(f\) and \(g\) determine the behavior and form the basis for evaluating the divisor of the rational function such that \ (\operatorname{div}(f/g) = \operatorname{div}(f) - \operatorname{div}(g)\).

Rational functions and their properties allow mathematicians to delve into and uncover the algebraic structure and symmetries within projective spaces like \mathbb{P}^1\.