The Commutativity of Pullback with Div

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Introduction

In the realm of algebraic geometry, particularly in the study of curves, the concept of pullback and divisors plays a crucial role. The pullback of a divisor under a morphism of curves is a fundamental operation that enables us to relate the geometry of different curves. In this article, we will delve into the properties of the pullback of a divisor and explore the commutativity of this operation with the divisor itself.

Background and Notations

Before we dive into the main discussion, let's establish some notations and background information. We assume that $K$ is a perfect field and $\overline{K}$ is its algebraic closure. Let $X$ and $Y$ be two curves over $K$ , and let $\phi: X \to Y$ be a morphism of curves. We denote the function field of $X$ by $K(X)$ and the function field of $Y$ by $K(Y)$ . A divisor on $X$ is a formal sum of prime divisors on $X$ , and we denote the group of divisors on $X$ by $\text{Div}(X)$ . The pullback of a divisor $D$ on $Y$ under $\phi$ is denoted by $\phi^*D$ .

The Pullback of a Divisor

The pullback of a divisor $D$ on $Y$ under $\phi$ is defined as follows:

\phi^*D = \sum_{P \in X} \phi^*(D_P) \cdot P

where $D_P$ is the local component of $D$ at the point $P \in Y$ . The pullback of a divisor is a divisor on $X$ , and it satisfies the following properties:

Linearity: The pullback of a sum of divisors is the sum of their pullbacks: $\phi^*(D_1 + D_2) = \phi^*D_1 + \phi^*D_2$ .
Multiplicativity: The pullback of a product of divisors is the product of their pullbacks: $\phi^*(D_1 \cdot D_2) = \phi^*D_1 \cdot \phi^*D_2$ .

The Commutativity of Pullback with Div

The main result of this article is the commutativity of the pullback of a divisor with the divisor itself. Specifically, we have:

Theorem 1. Let $X$ and $Y$ be two curves over $K$ , and let $\phi: X \to Y$ be a morphism of curves. Let $D$ be a divisor on $Y$ . Then, we have:

\phi^*D = D \circ \phi

where $D \circ \phi$ is the composition of the divisor $D$ with the morphism $\phi$ .

Proof. Let $P \in X$ be a point, and let $Q = \phi(P) \in Y$ be the corresponding point on $Y$ . We have:

\phi^*D_P = \phi^*(D_Q) = D_Q \circ \phi

where $D_P$ and $D_Q$ are the local components of $D$ at the points $P$ and $Q$ , respectively. Therefore, we have:

\phi^*D = \sum_{P \in X} \phi^*(D_P) \cdot P = \sum_{P \in X} D_Q \circ \phi \cdot P = D_Q \circ \phi = D \circ \phi

This completes the proof of Theorem 1.

Applications and Examples

The commutativity of the pullback of a divisor with the divisor itself has several important applications in algebraic geometry. For example, it enables us to relate the geometry of different curves and to study the properties of divisors on curves. Here are a few examples:

Pullback of a divisor under a finite morphism: Let $\phi: X \to Y$ be a finite morphism of curves. Then, the pullback of a divisor $D$ on $Y$ under $\phi$ is a divisor on $X$ that is equal to the composition of $D$ with $\phi$ .
Pullback of a divisor under a rational map: Let $\phi: X \to Y$ be a rational map of curves. Then, the pullback of a divisor $D$ on $Y$ under $\phi$ is a divisor on $X$ that is equal to the composition of $D$ with $\phi$ .

Conclusion

In this article, we have explored the properties of the pullback of a divisor under a morphism of curves. We have shown that the pullback of a divisor commutes with the divisor itself, and we have provided several examples and applications of this result. The commutativity of the pullback of a divisor with the divisor itself is a fundamental property of algebraic geometry that has far-reaching implications for the study of curves and divisors.

References

The Arithmetic of Elliptic Curves, second edition by Joseph H. Silverman. Springer-Verlag, 2009.
Algebraic Geometry, Volume 1 by Robin Hartshorne. Springer-Verlag, 1977.

Introduction

In our previous article, we explored the properties of the pullback of a divisor under a morphism of curves. We showed that the pullback of a divisor commutes with the divisor itself, and we provided several examples and applications of this result. In this article, we will answer some of the most frequently asked questions about the pullback of a divisor and its commutativity with the divisor itself.

Q: What is the pullback of a divisor?

A: The pullback of a divisor $D$ on a curve $Y$ under a morphism $\phi: X \to Y$ is a divisor on $X$ that is defined as follows:

\phi^*D = \sum_{P \in X} \phi^*(D_P) \cdot P

where $D_P$ is the local component of $D$ at the point $P \in Y$ .

Q: What is the commutativity of the pullback of a divisor with the divisor itself?

A: The commutativity of the pullback of a divisor with the divisor itself states that for any divisor $D$ on a curve $Y$ and any morphism $\phi: X \to Y$ , we have:

\phi^*D = D \circ \phi

where $D \circ \phi$ is the composition of the divisor $D$ with the morphism $\phi$ .

Q: What are some examples of the commutativity of the pullback of a divisor with the divisor itself?

A: Here are a few examples:

Pullback of a divisor under a finite morphism: Let $\phi: X \to Y$ be a finite morphism of curves. Then, the pullback of a divisor $D$ on $Y$ under $\phi$ is a divisor on $X$ that is equal to the composition of $D$ with $\phi$ .
Pullback of a divisor under a rational map: Let $\phi: X \to Y$ be a rational map of curves. Then, the pullback of a divisor $D$ on $Y$ under $\phi$ is a divisor on $X$ that is equal to the composition of $D$ with $\phi$ .

Q: What are some applications of the commutativity of the pullback of a divisor with the divisor itself?

A: Here are a few examples:

Relating the geometry of different curves: The commutativity of the pullback of a divisor with the divisor itself enables us to relate the geometry of different curves and to study the properties of divisors on curves.
Studying the properties of divisors on curves: The commutativity of the pullback of a divisor with the divisor itself is a fundamental property of algebraic geometry that has far-reaching implications for the study of curves and divisors.

Q: What are some further reading resources on the topic of algebraic geometry and curves?

A: Here are a few resources:

Algebraic Geometry, Volume 2 by Robin Hartshorne. Springer-Verlag, 1977.
The Geometry of Algebraic Curves, Volume 1 by Enrico Arbarello, Corrado De Concini, Chiara Giacaglia, and Marino Palleschi. Springer-Verlag, 2011.
The Geometry of Algebraic Curves, Volume 2 by Enrico Arbarello, Corrado De Concini, Chiara Giacaglia, and Marino Palleschi. Springer-Verlag, 2011.

Conclusion

In this article, we have answered some of the most frequently asked questions about the pullback of a divisor and its commutativity with the divisor itself. We hope that this article has provided a helpful resource for those interested in the topic of algebraic geometry and curves.

References

The Arithmetic of Elliptic Curves, second edition by Joseph H. Silverman. Springer-Verlag, 2009.
Algebraic Geometry, Volume 1 by Robin Hartshorne. Springer-Verlag, 1977.
The Geometry of Algebraic Curves, Volume 1 by Enrico Arbarello, Corrado De Concini, Chiara Giacaglia, and Marino Palleschi. Springer-Verlag, 2011.
The Geometry of Algebraic Curves, Volume 2 by Enrico Arbarello, Corrado De Concini, Chiara Giacaglia, and Marino Palleschi. Springer-Verlag, 2011.