Chapter 8: Problem 19
If \(f: C \rightarrow V\) is a morphism from a projective curve to a variety \(V\), then \(f(C)\) is a closed subvariety of \(V\). (Hint: Consider \(C^{\prime}=\) closure of \(f(C)\) in \(V\).)
Short Answer
Expert verified
Question: Prove that if we have a morphism \(f: C \rightarrow V\), where \(C\) is a projective curve, then the image of \(C\) under \(f\), denoted by \(f(C)\), is a closed subvariety of \(V\).
Answer: By considering the closure of the image, \(C^{\prime}\), and the restricted morphism, we showed that \(f(C)\) is a closed subvariety of \(C^{\prime}\) and, therefore, it must also be a closed subvariety of \(V\).
Step by step solution
01
Definition of closed subvariety
In order to show that \(f(C)\) is a closed subvariety of \(V\), we need to recall what a closed subvariety is. A subvariety \(Y\) of a variety \(X\) is closed if it is the zero set of a collection of polynomials in the coordinate ring of \(X\), that is, \(Y = V(I)\) for some ideal \(I\) of the coordinate ring \(R_x = k[x_1, \dots, x_n]\).
02
Definition of the closure of a set
The closure of a set \(S\) in a topological space is the intersection of all closed sets containing \(S\). In algebraic geometry, this is defined in terms of the Zariski topology: a set is closed in the Zariski topology if it is the zero set of a collection of polynomials. The closure of \(S\) in a variety \(V\) will be denoted as \(\overline{S}\).
03
Morphism preserves the projective curve condition
A morphism, by definition, preserves the algebraic structure of its domain and its codomain. In particular, if \(C\) is a projective curve, then its image under a morphism will inherit this property. Therefore, \(f(C)\) is also a projective curve (possibly degenerate).
04
Consider the closure of the image
As suggested by the hint, let's consider the closure of \(f(C)\) in \(V\), denoted by \(C^{\prime}\). Since \(f(C)\) is a projective curve, and \(C^{\prime}\) is the smallest closed set containing \(f(C)\), it follows that \(C^{\prime}\) must also be a projective curve.
05
Restricted morphism
Now let's consider the morphism \(f\) restricted to its image \(f(C)\). We can define a new morphism \(g: C \rightarrow C^{\prime}\), where \(g = f|_{C}\). Since \(g\) is a morphism between projective curves, it must preserve the algebraic structure, and thus \(f(C)\) must be a closed subvariety of \(C^{\prime}\).
06
Showing image is a closed subvariety
Now let's recall that \(C^{\prime}\) is the smallest closed set containing \(f(C)\). From the previous step, it follows that \(f(C)\) is a closed subvariety of \(C^{\prime}\). Hence, \(f(C)\) must be a closed subvariety of \(V\) as well, since being closed is a transitive property.
By showing that \(f(C)\) is a closed subvariety of \(V\), we've completed the exercise.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Algebraic Geometry
Algebraic geometry is a branch of mathematics that combines algebra, which deals with equations and their solutions, with geometry, which studies shapes and their properties. Imagine it as a blending of numbers and shapes, where equations represent geometric objects called 'varieties.' These varieties are the solutions to polynomial equations and can take various forms such as curves, surfaces, or more complex shapes in higher dimensions.
In algebraic geometry, we often work with concepts like 'closed subvarieties,' which are subsets of varieties that are defined by polynomial equations. These subvarieties are 'closed' in a certain sense that we'll see soon when discussing the Zariski topology. The main goal in this field is to study the properties and structures of these geometric objects that arise from algebraic equations.
In algebraic geometry, we often work with concepts like 'closed subvarieties,' which are subsets of varieties that are defined by polynomial equations. These subvarieties are 'closed' in a certain sense that we'll see soon when discussing the Zariski topology. The main goal in this field is to study the properties and structures of these geometric objects that arise from algebraic equations.
Projective Curve
A projective curve is a one-dimensional variety in projective space. Unlike curves in our usual three-dimensional space, projective curves have some special properties. For example, two distinct lines in projective space will always intersect at a point (think of this as a way to solve the classic parallel lines problem, as all lines 'meet at infinity'). This 'projective' nature of the space makes it very useful for studying geometric properties that don't depend on the specific coordinates chosen.
Projective curves are important because they provide a setting to work with polynomial equations that may not have solutions in the standard sense, like those that go off to 'infinity.' In algebraic geometry, working with projective curves helps us understand not just their shape and structure, but also how they interact with other projective objects through morphisms.
Projective curves are important because they provide a setting to work with polynomial equations that may not have solutions in the standard sense, like those that go off to 'infinity.' In algebraic geometry, working with projective curves helps us understand not just their shape and structure, but also how they interact with other projective objects through morphisms.
Zariski Topology
Zariski topology is a unique way of thinking about closeness and openness that is very different from what we're used to with our ordinary, 'Euclidean' intuition. In the Zariski topology, a set is considered 'closed' if it can be described as the set of all solutions to a given set of polynomial equations. This concept of topology might seem odd at first because large sets, such as the complement of a single point, are considered to be 'open.'
The reason why this is so handy in algebraic geometry is that it aligns perfectly with our algebraic structures. It gives us a way to talk about geometric concepts like continuity and closure in the context of varieties and their subvarieties. The closure of a set becomes a crucial idea because it allows us to consider all the possible 'limits' of that set under polynomial equations. It's like casting a net that captures all potential solutions, some of which could be at 'infinity.'
The reason why this is so handy in algebraic geometry is that it aligns perfectly with our algebraic structures. It gives us a way to talk about geometric concepts like continuity and closure in the context of varieties and their subvarieties. The closure of a set becomes a crucial idea because it allows us to consider all the possible 'limits' of that set under polynomial equations. It's like casting a net that captures all potential solutions, some of which could be at 'infinity.'
Morphism
In algebraic geometry, a morphism is a map between geometric objects that respects their algebraic structure. Think of it like a translator between languages that not only converts words but ensures the whole sentence keeps its meaning. A morphism takes points from one variety to another in such a way that the equations defining the varieties are preserved.
Why is this so special? Because morphisms allow us to study how different varieties relate to each other and how their properties may be transferred or transformed. Our exercise deals with a morphism from a projective curve to a variety, a situation ripe with implications for the shape and nature of the curve's image within the larger variety. By understanding the behavior of morphisms, we can start to unravel the intricate tapestry that is the geometry of polynomial solutions.
Why is this so special? Because morphisms allow us to study how different varieties relate to each other and how their properties may be transferred or transformed. Our exercise deals with a morphism from a projective curve to a variety, a situation ripe with implications for the shape and nature of the curve's image within the larger variety. By understanding the behavior of morphisms, we can start to unravel the intricate tapestry that is the geometry of polynomial solutions.