Question

Let \(C\) be a projective curve, \(P \in C .\) Then there is a birational morphism \(f: C \rightarrow\) \(C^{\prime}, C^{\prime}\) a projective plane curve, such that \(f^{-1}(f(P))=\\{P\\} .\) We outline a proof: (a) We can assume: \(C \subset \mathbb{P}^{n+1}\) Let \(T, X_{1}, \ldots, X_{n}, Z\) be coordinates for \(\mathbb{P}^{n+1}\); Then \(C \cap V(T)\) is finite; \(C \cap V(T, Z)=\varnothing ; P=[0: \ldots: 0: 1] ;\) and \(k(C)\) is algebraic over \(k(u)\) where \(u=\bar{T} / \bar{Z} \in k(C)\). (b) For each \(\lambda=\left(\lambda_{1}, \ldots, \lambda_{n}\right) \in k^{n}\), let \(\varphi_{\lambda}: C \rightarrow \mathbb{P}^{2}\) be defined by the formula \(\varphi([t:\) \(\left.\left.x_{1}: \ldots: x_{n}: z\right]\right)=\left[t: \sum \lambda_{i} x_{i}: z\right] .\) Then \(\varphi_{\lambda}\) is a well-defined morphism, and \(\varphi_{\lambda}(P)=\) \([0: 0: 1] .\) Let \(C^{\prime}\) be the closure of \(\varphi_{\lambda}(C)\) (c) The variable \(\lambda\) can be chosen so \(\varphi_{\lambda}\) is a birational morphism from \(C\) to \(C^{\prime}\), and \(\varphi_{\lambda}^{-1}([0: 0: 1])=\\{P\\} .\) (Use Problem \(6.32\) and the fact that \(C \cap V(T)\) is finite).

Short answer

Question: Prove the existence of a birational morphism between a projective curve C and a plane curve C' in the given conditions. Solution: We have three main parts in our proof: (a) setting up the necessary conditions and assumptions, (b) defining the morphism φ_lambda, and (c) showing the existence of a suitable lambda such that φ_lambda has the desired properties. (a) The given problem assumes that C is in the projective space \(\mathbb{P}^{n+1}\) with coordinates T, \(X_1, \ldots, X_n, Z\), and C intersects the variety \(V(T)\) in a finite set of points, C does not intersect the variety \(V(T,Z)\), and point P has the coordinates \([0: \ldots: 0: 1]\). (b) We define the morphism \(\varphi_\lambda: C \rightarrow \mathbb{P}^2\) for each vector \(\lambda = (\lambda_1, \ldots, \lambda_n) \in k^n\) by \(\varphi_\lambda([t: x_1: \ldots: x_n: z]) = [t: \sum \lambda_i x_i: z]\). We let \(C'\) be the closure of the image of C under \(\varphi_\lambda\). (c) To find a λ such that the morphism φ_lambda is birational and its inverse image of \([0: 0: 1]\) is the singleton set containing P, we can use the result of Problem 6.32 and the given fact that C intersects V(T) finitely. There exists a λ for which \(P \notin B\), and for this λ, the morphism φ_lambda is birational, and the inverse image of \([0: 0: 1]\) under φ_lambda is the singleton set containing P. Hence, the existence of such a λ establishes the existence of a birational morphism between C and C'.

Step-by-step solution

Part (a): Assumptions and conditions

First, we make some assumptions about our curve C and point P. The given exercise assumes that C is in the projective space \(\mathbb{P}^{n+1}\) with coordinates T, \(X_1, \ldots, X_n, Z\). Furthermore, we assume that C intersects the variety \(V(T)\) in a finite set of points, C does not intersect the variety \(V(T,Z)\), and P has the coordinates \([0: \ldots: 0: 1]\). The algebraic closure of the function field \(k(C)\) is assumed to be algebraic over the field \(k(u)\) where \(u = \bar{T}/\bar{Z} \in k(C)\).

Part (b): Defining the morphism φ_lambda

For each vector \(\lambda = (\lambda_1, \ldots, \lambda_n) \in k^n\), we define the morphism \(\varphi_\lambda: C \rightarrow \mathbb{P}^2\) by the formula \(\varphi_\lambda([t: x_1: \ldots: x_n: z]) = [t: \sum \lambda_i x_i: z]\). This is a well-defined morphism and the image of point P under this morphism is \([0: 0: 1]\). We let \(C'\) be the closure of the image of C under \(\varphi_\lambda\).

Part (c): Existence of suitable lambda

Now, we want to find a λ such that the morphism φ_lambda is birational and its inverse image of \([0: 0: 1]\) is the singleton set containing P. To do this, we can use the result of Problem 6.32 and the given fact that C intersects V(T) finitely. From Problem 6.32, we know that there exists a finite subset \(B \subset C\) such that for any point \(Q \in C\) not in B, the morphism \(\varphi_\lambda\) restricted to \(C - B\) is an open immersion. Since C intersects V(T) in a finite set of points, there is at least one λ for which \(P \notin B\). For this λ, the morphism φ_lambda is birational, and since the image of P is \([0: 0: 1]\), the inverse image of \([0: 0: 1]\) under φ_lambda is the singleton set containing P. Thus, the existence of such λ is established.

Key concepts

Projective Curve

Imagine drawing a loop or a curve on a piece of paper, so smooth that it never stops or has gaps. This is akin to what a **projective curve** represents in the realm of algebraic geometry. Defined over the projective space, these curves are an essential structure for understanding more complex geometric forms.

In mathematical terms, a projective curve is characterized as a one-dimensional closed subset of a projective space like \(\mathbb{P}^{n+1}\). This space is slightly different from the basic Euclidean plane because it adds new points at infinity, turning lines that never meet on a flat plane into those that do interact somewhere at infinity.

A crucial feature of projective curves is that they can be described by polynomial equations. These equations don't change even as you apply various transformations, making them very predictable and essential in studying birational morphisms. Essentially, these curves enable mathematicians to transform geometric shapes into algebraic expressions suitable for analysis.

Algebraic Geometry

Diving into algebraic geometry is akin to blending two major branches of mathematics: algebra and geometry. It utilizes the rigorous logical structure of algebra to answer questions about geometry.

- **Basic Foundation**: At its core, algebraic geometry involves studying zero sets of polynomials. That is, exploring the shapes and spaces defined by algebraic equations.
- **Algebraic Structures**: These include points, lines, and curves. Interestingly, algebraic geometry doesn’t just deal with numbers; it represents geometric entities in terms of algebraic functions.

The intersection of algebra and geometry offers rich insights, explaining not only the shapes and sizes but also the relationships between different geometric entities. For example, algebraic geometry helps us analyze projective curves through polynomial expressions. Moreover, algebraic geometry allows birational transformations, which simplify terms while maintaining a shape's intrinsic properties, which is tremendously useful in complex geometric problem-solving.

Function Field

When studying curves through their function fields, think of it as viewing all the possible values a function can process and output. This concept is pivotal in understanding the arithmetic and intricacies of algebraic curves.

A **function field** is associated with an algebraic variety, like a projective curve, and consists of ratios of polynomials, known as rational functions. For any projective curve \(C\), the function field \(k(C)\) aids in describing the functions defined on \(C\) in an algebraic way.

Understanding function fields simplifies algebraic geometry problems by transforming geometric challenges into algebraic operations. For instance, by considering an algebraic curve as a projective curve in a function field, it becomes possible to explore its algebraic closure over a smaller base field — thus making it easier to handle complex equations and structures.

Projective Space

To grasp the concept of projective space, think of a realm where common parallel lines meet. Unlike your regular plane, this space connects dots at infinity, offering a broader perspective on geometric configurations.

Projective space, often denoted as \(\mathbb{P}^n\), provides a framework where every line indeed intersects, eliminating the notion of parallelism. This space is crucial in advanced geometry for its ability to simplify the study of curves and surfaces by treating them uniformly.

- **Coordinates in Projective Space**: Represented using homogeneous coordinates, every point’s position uses an extra dimension to codify the "directions to infinity."
- **Role in Geometry and Algebra**: This environment transforms algebraic problems into geometric ones, allowing for better manipulation and simplification of algebraic forms through birational transformations.

Projective space gives researchers the tools to not only study geometric constraints but also to address complex curves and intersections, making it indispensable in algebraic geometry.