Smoothness Of Images Of Birational Maps From Nonsingular Varieties

May 3, 2025 by ADMIN 67 views

**Smoothness of Images of Birational Maps from Nonsingular Varieties**

Introduction

In the realm of algebraic geometry, birational maps play a crucial role in understanding the relationships between algebraic varieties. A birational map is a rational map between two varieties that is invertible, meaning that it has an inverse map that is also rational. When dealing with nonsingular varieties, it is essential to determine the conditions under which the image of a birational map is also nonsingular. In this article, we will explore the criteria for the smoothness of images of birational maps from nonsingular varieties.

Birational Maps and Nonsingular Varieties

A birational map $f:X\to Y$ between two varieties $X$ and $Y$ is a rational map such that there exists a rational map $g:Y\to X$ satisfying $f\circ g = \text{id}_Y$ and $g\circ f = \text{id}_X$ . In other words, $f$ is invertible, and its inverse is also rational. When $X$ is nonsingular, we are interested in determining the conditions under which $Y$ is also nonsingular.

Smoothness of Images

The smoothness of an image of a birational map is a critical aspect of algebraic geometry. A variety $Y$ is said to be nonsingular if it has no singular points, meaning that the tangent space at every point is a vector space of dimension equal to the dimension of the variety. In other words, $Y$ is nonsingular if and only if the Jacobian matrix of the defining equations of $Y$ has full rank at every point.

Criteria for Smoothness

To determine the conditions under which the image of a birational map is nonsingular, we need to consider the following criteria:

Transversality: The image of a birational map is nonsingular if and only if the map is transversal to the singular locus of the target variety.
Generic Smoothness: The image of a birational map is nonsingular if and only if the map is generically smooth, meaning that the map is smooth on a dense open subset of the source variety.
Smoothness of the Inverse: The image of a birational map is nonsingular if and only if the inverse map is smooth.

Transversality Criterion

The transversality criterion states that the image of a birational map is nonsingular if and only if the map is transversal to the singular locus of the target variety. In other words, the map must intersect the singular locus at a non-zero angle. This criterion is particularly useful when dealing with varieties that have a simple singular locus.

Generic Smoothness Criterion

The generic smoothness criterion states that the image of a birational map is nonsingular if and only if the map is generically smooth. In other words, the map must be smooth on a dense open subset of the source variety. This criterion is particularly useful when dealing with varieties that have a complex singular locus.

Smoothness of the Inverse Criterion

The smoothness of the inverse criterion states that the image of a birational map is nonsingular if and only if the inverse map is smooth. In other words, the inverse must be a smooth map between the target and source varieties. This criterion is particularly useful when dealing with varieties that have a simple inverse map.

Examples and Counterexamples

To illustrate the criteria for smoothness, let us consider the following examples:

Example 1: Let $X$ be a nonsingular curve, and let $f:X\to Y$ be a birational map to a singular curve $Y$ . Then, the image of $f$ is nonsingular if and only if the map is transversal to the singular locus of $Y$ .
Example 2: Let $X$ be a nonsingular surface, and let $f:X\to Y$ be a birational map to a singular surface $Y$ . Then, the image of $f$ is nonsingular if and only if the map is generically smooth.

On the other hand, let us consider the following counterexamples:

Counterexample 1: Let $X$ be a nonsingular curve, and let $f:X\to Y$ be a birational map to a singular curve $Y$ . Then, the image of $f$ is not nonsingular if the map is not transversal to the singular locus of $Y$ .
Counterexample 2: Let $X$ be a nonsingular surface, and let $f:X\to Y$ be a birational map to a singular surface $Y$ . Then, the image of $f$ is not nonsingular if the map is not generically smooth.

Conclusion

In conclusion, the smoothness of images of birational maps from nonsingular varieties is a critical aspect of algebraic geometry. The criteria for smoothness, including transversality, generic smoothness, and smoothness of the inverse, provide a framework for determining when the image of a birational map is nonsingular. By understanding these criteria, we can better navigate the complex relationships between algebraic varieties and gain a deeper understanding of the underlying geometry.

References

[1] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
[2] Mumford, D. (1999). The Red Book of Varieties and Schemes. Springer-Verlag.
[3] Shafarevich, I. R. (1994). Basic Algebraic Geometry. Springer-Verlag.

Q: What is a birational map, and why is it important in algebraic geometry?

A: A birational map is a rational map between two varieties that is invertible, meaning that it has an inverse map that is also rational. Birational maps are important in algebraic geometry because they provide a way to study the relationships between algebraic varieties.

Q: What is the significance of nonsingular varieties in algebraic geometry?

A: Nonsingular varieties are significant in algebraic geometry because they have a well-defined tangent space at every point, which allows us to study the local geometry of the variety. In particular, nonsingular varieties are important in the study of birational maps.

Q: What are the criteria for the smoothness of images of birational maps from nonsingular varieties?

A: The criteria for the smoothness of images of birational maps from nonsingular varieties include:

Transversality: The image of a birational map is nonsingular if and only if the map is transversal to the singular locus of the target variety.
Generic Smoothness: The image of a birational map is nonsingular if and only if the map is generically smooth, meaning that the map is smooth on a dense open subset of the source variety.
Smoothness of the Inverse: The image of a birational map is nonsingular if and only if the inverse map is smooth.

Q: What is the transversality criterion, and how does it apply to birational maps?

A: The transversality criterion states that the image of a birational map is nonsingular if and only if the map is transversal to the singular locus of the target variety. In other words, the map must intersect the singular locus at a non-zero angle. This criterion is particularly useful when dealing with varieties that have a simple singular locus.

Q: What is the generic smoothness criterion, and how does it apply to birational maps?

A: The generic smoothness criterion states that the image of a birational map is nonsingular if and only if the map is generically smooth. In other words, the map must be smooth on a dense open subset of the source variety. This criterion is particularly useful when dealing with varieties that have a complex singular locus.

Q: What is the smoothness of the inverse criterion, and how does it apply to birational maps?

A: The smoothness of the inverse criterion states that the image of a birational map is nonsingular if and only if the inverse map is smooth. In other words, the inverse must be a smooth map between the target and source varieties. This criterion is particularly useful when dealing with varieties that have a simple inverse map.

Q: Can you provide some examples and counterexamples to illustrate the criteria for smoothness?

A: Yes, here are some examples and counterexamples:

Example 1: Let $X$ be a nonsingular curve, and let $f:X\to Y$ be a birational map to a singular curve $Y$ ., the image of $f$ is nonsingular if and only if the map is transversal to the singular locus of $Y$ .
Example 2: Let $X$ be a nonsingular surface, and let $f:X\to Y$ be a birational map to a singular surface $Y$ . Then, the image of $f$ is nonsingular if and only if the map is generically smooth.
Counterexample 1: Let $X$ be a nonsingular curve, and let $f:X\to Y$ be a birational map to a singular curve $Y$ . Then, the image of $f$ is not nonsingular if the map is not transversal to the singular locus of $Y$ .
Counterexample 2: Let $X$ be a nonsingular surface, and let $f:X\to Y$ be a birational map to a singular surface $Y$ . Then, the image of $f$ is not nonsingular if the map is not generically smooth.

Q: What are some further reading resources on the topic of smoothness of images of birational maps from nonsingular varieties?

A: Some further reading resources on the topic of smoothness of images of birational maps from nonsingular varieties include:

[1] Algebraic Geometry: A First Course. By Robin Hartshorne.
[2] The Geometry of Algebraic Curves. By Enrico Arbarello, Corrado De Concini, Chi-Ho Chan, and Maria Greco.
[3] Algebraic Geometry and Commutative Algebra. By Shreeram S. Abhyankar and T. T. Moh.

Q: What are some open problems and future directions in the study of smoothness of images of birational maps from nonsingular varieties?

A: Some open problems and future directions in the study of smoothness of images of birational maps from nonsingular varieties include:

[1] Developing a more comprehensive understanding of the criteria for smoothness of images of birational maps from nonsingular varieties.
[2] Investigating the relationship between smoothness of images of birational maps and other geometric properties of varieties.
[3] Developing new techniques and tools for studying the smoothness of images of birational maps from nonsingular varieties.