Question

Show that a morphism from a projective curve \(X\) to a curve \(Y\) is either constant or surjective; if it is surjective, \(Y\) must be projective.

Short answer

Question: Show that a morphism from a projective curve to another curve is either constant or surjective, and if it is surjective, the curve must also be projective. Answer: A morphism from a projective curve to another curve is either constant, meaning it maps every point on the domain curve to a single point in the codomain curve, or surjective, meaning it covers every point of the codomain curve with at least one point from the domain curve. If the morphism is surjective, it implies that for any two points in the codomain curve, they have a common point, which is a property of projective curves, thus making the curve projective.

Step-by-step solution

Understand projective curves

A projective curve is a curve that can be embedded in a projective space. Projective spaces have the property that every two curves intersect at some point, even if they appear to be parallel in the affine space of the same dimension. In other words, projective curves have no missing points or points at infinity.

Understand constant and surjective morphisms

A constant morphism is a morphism that maps every point on the domain curve to the same point in the codomain curve. A surjective morphism is a morphism that covers every point of the codomain curve with at least one point from the domain curve. In this problem, we want to show that a morphism from a projective curve to another curve must be either constant or surjective.

Show that the morphism is either constant or surjective

Consider a morphism \(f: X \rightarrow Y\) from a projective curve \(X\) to a curve \(Y\). If the morphism is constant, then every point on \(X\) is mapped to a single point on \(Y\). This means that all points on the projective curve \(X\) collapse to a single point, and the image of \(f\) is a single point in \(Y\). This satisfies the condition that the morphism is constant. Now, if the morphism is not constant, then there exists at least two distinct points on \(X\) that have distinct images in \(Y\). Since \(X\) is projective, any two points on \(X\) must have a common point. Call this common point \(P\). Now, consider any point \(Q\) on \(Y\). Since \(f\) is not constant, there must be some point on \(X\) that is mapped to \(Q\) by \(f\). Therefore, the morphism \(f\) covers every point of \(Y\) with at least one point from \(X\), which means that the morphism is surjective. Hence, the morphism from a projective curve \(X\) to a curve \(Y\) is either constant or surjective.

Show that if the morphism is surjective, \(Y\) must be projective

Suppose that \(f: X \rightarrow Y\) is a surjective morphism from a projective curve \(X\) to a curve \(Y\). Let \(A\) and \(B\) be any two points in \(Y\). Since the morphism is surjective, there exist points \(P\) and \(Q\) in \(X\) such that \(f(P) = A\) and \(f(Q) = B\). Since \(X\) is projective, there exists a point \(R\in X\) such that \(P\) and \(Q\) intersect at \(R\). Now, applying the morphism \(f\), we have \(A=f(P)\) and \(B=f(Q)\), and \(R\) must be mapped to a common point in \(Y\), call it \(C\). Hence, any two points in \(Y\) have a common point \(C\). This shows that \(Y\) also has the property that every two points have a common point, which is the property of a projective curve. Therefore, if the morphism from a projective curve \(X\) to a curve \(Y\) is surjective, then \(Y\) must be projective.

Key concepts

Morphisms

In algebraic geometry, morphisms are essential tools for understanding the relationships between various geometric structures. A morphism, simply put, is a map between two algebraic varieties that preserves the structure of the varieties. These maps can bridge gaps between seemingly unrelated geometrical shapes and structures.

Morphisms in the context of curves, particularly projective curves, help us explore how one curve can be transformed or related to another. For example, if we take a morphism from a projective curve to another curve, it provides a way to compare and study their properties. This morphism can either bring the entire curve to a single point (constant) or reach every point of the other curve (surjective).

Understanding morphisms is crucial to navigating the landscape of algebraic geometry, as they enable the transformation and study of complex structures in a meaningful way.

Surjective Maps

Surjective maps, also known as onto maps, are special types of morphisms with an important property: every point in the target space is mapped by at least one point in the domain space. Essentially, if you have a surjective map from curve X to curve Y, this means that every point in curve Y corresponds to some point or points on curve X.

Surjectivity is significant in algebraic geometry because it ensures the coverage of one space by another. Hence, in the context of projective curves, if the morphism is surjective, the image under this morphism is the entire target curve. This characteristic makes surjective morphisms incredibly useful when trying to understand mappings between curves, especially when determining if a curve like Y inherits certain properties from a projective curve X.

• Surjective morphisms show that every point in Y has a preimage in X.
• This is a foundational concept when demonstrating that Y itself needs to maintain certain structural integrity, such as being projective itself if X is projective.

Constant Morphisms

Constant morphisms are quite straightforward — they map every point of a domain space, such as a projective curve, to a single point in the codomain. In other words, no matter where you start on the initial curve, you will always end up at the same spot on the target curve.

These morphisms become especially relevant in the context of projective curves due to their simplicity and inherent properties. For a morphism from one projective curve X to another curve Y to be considered constant, every point on X must map to the same single point on Y.

• Constant morphisms imply that the entire structure of X collapses to a point in Y.
• This scenario demonstrates one extreme possibility when analyzing morphisms between curves, the other being surjectivity.

Algebraic Geometry

Algebraic geometry is a branch of mathematics where geometry and algebra come together. This field primarily deals with solving equations and understanding the shapes that these solutions form. At its core, algebraic geometry uses algebraic techniques to tackle geometrical problems, giving us a more profound understanding of various geometric spaces, including curves, surfaces, and higher-dimensional varieties.

Projective curves, a key focus in algebraic geometry, are understood and analyzed through the lens of morphisms. They are deeply connected to the structure of these curves and their embeddings into projective spaces, where they acquire interesting properties, like intersecting at points.

• The study of algebraic geometry involves examining the properties and relationships of projective curves.
• Morphisms and maps such as constant and surjective are central to this field, offering insights into how different curves relate to one another.

Through algebraic geometry, complex relationships between curves can be simplified and understood, making it a powerful tool in the study of geometric structures.