Question

\(\mathbb{P}^{n} \times \mathbb{P}^{m}\) is birationally equivalent to \(\mathbb{P}^{n+m}\). Show that \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) is not isomorphic to \(\mathbb{P}^{2}\). (Hint: \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) has closed subvarieties of dimension one that do not intersect.)

Short answer

In order to prove that \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) is not isomorphic to \(\mathbb{P}^{2}\), we introduced two closed subvarieties of dimension one in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\), \(V_1\) and \(V_2\), which have empty intersection. Whereas in \(\mathbb{P}^2\), any two distinct curves (one-dimensional subvarieties) necessarily intersect in at least one point. This contrasting property led us to the conclusion that \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) is not isomorphic to \(\mathbb{P}^{2}\).

Step-by-step solution

Introduce suitable subvarieties of \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

Consider the subvarieties \(V_1=\{([x_1:x_2],[y_1:y_2]) \in \mathbb{P}^1\times \mathbb{P}^1 \mid x_2=0\}\) and \(V_2=\{([x_1:x_2],[y_1:y_2]) \in \mathbb{P}^1\times \mathbb{P}^1 \mid y_1=0\}\). Both subvarieties are closed and have dimension one, as they are isomorphic to \(\mathbb{P}^1\).

Show that the subvarieties do not intersect

Observe that any point in the intersection \(V_1 \cap V_2\) would satisfy \(x_2 = 0\) and \(y_1=0\). This means that such a point would be of the form \(([x_1:0],[0:y_2])\). However, since neither \(x_1\) nor \(y_2\) can simultaneously be zero, the intersection \(V_1 \cap V_2\) is empty.

Count the curves in \(\mathbb{P}^{2}\) and compare with \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

In \(\mathbb{P}^2\), any two distinct curves (one-dimensional subvarieties) necessarily intersect in at least one point. This is due to the well-known Bezout's theorem in algebraic geometry, which states that for two algebraic curves in \(\mathbb{P}^2\) with degrees \(m\) and \(n\), their intersection points sum up to \(m \cdot n\) (counted with multiplicities), provided that they do not have a common component. Since the product of their degrees is nonzero for any two distinct curves, it implies that any two curves in \(\mathbb{P}^2\) intersect.

Conclude that \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) is not isomorphic to \(\mathbb{P}^{2}\)

As shown in step 2, we found two non-intersecting subvarieties of dimension one in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). However, in step 3, we showed that such a property does not hold for \(\mathbb{P}^2\), where any two curves always intersect. Due to this contrasting property, we conclude that \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) is not isomorphic to \(\mathbb{P}^{2}\).

Key concepts

Birational Equivalence

Understanding birational equivalence is essential in comprehending the relationship between algebraic varieties. Birational equivalence is a type of relationship between two algebraic varieties where, roughly speaking, each variety can be morphed into the other by a sequence of rational maps that have inverses which are also rational. This concept is a bit like having two distinct puzzles that can be rearranged into each other without any pieces leftover.

Specifically, two algebraic varieties are birationally equivalent if there exist rational functions that map one variety to the other and vice versa, where these functions behave nicely, i.e., they are isomorphisms when restricted to some open subset of each variety. Birational equivalence implies that the varieties share many properties and, intuitively, they look 'the same' except maybe on sets of smaller dimension where the rational maps are not defined.

In our context, although \(\mathbb{P}^{n} \times \mathbb{P}^{m}\) is birationally equivalent to \(\mathbb{P}^{n+m}\), the problem shows that being birationally equivalent does not necessarily mean that two varieties are isomorphic, meaning there isn't a one-to-one correspondence that preserves the structure of the varieties everywhere.

Projective Space

Projective space is a fundamental concept in algebraic geometry. It is denoted by \(\mathbb{P}^{n}\) for any non-negative integer \(n\), representing the set of lines through the origin in \(\mathbb{R}^{n+1}\), except we consider all points on a line to be equivalent. You can think of projective space as an extension of our usual Euclidean space, where we include 'points at infinity' to neatly tie up the loose ends - where parallel lines meet and curves close up.

One powerful feature of projective space is how it simplifies the treatment of polynomial equations, allowing us to handle complex properties of curves and surfaces, such as intersections, by considering their projective counterparts. In the context of our exercise, the product \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) and \(\mathbb{P}^{2}\) might seem similar at first glance but are distinct in their projective geometry, specifically in how subvarieties interact within these spaces.

Bezout's Theorem

Bezout's theorem is a cornerstone of algebraic geometry, touching on the beautiful intersection theory within projective spaces. At its heart, the theorem states that the number of intersection points of two algebraic curves in the projective plane \(\mathbb{P}^{2}\) is equal to the product of their degrees, assuming the curves are without common components and counted with proper multiplicity.

This theorem allows us to make precise predictions about how curves in the projective plane should behave. For instance, two lines (each of degree one) should intersect in one point since \(1\times1=1\). In more complex situations, like a line and a parabola (degree two), there should be two intersection points, which could be real, complex, or one point counted twice if the line is tangent to the parabola.

Referring to our problem, Bezout's theorem directly contradicts the situation in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) where we find non-intersecting subvarieties, which simply cannot happen in \(\mathbb{P}^{2}\) because of Bezout’s insight.

Algebraic Subvarieties

Algebraic subvarieties are subsets of algebraic varieties that are, in themselves, algebraic varieties. Imagine an algebraic variety as a shape or a surface, and subvarieties as smaller shapes or curves lying on the surface, all defined by polynomial equations. In projective spaces, these subvarieties can be points, lines, curves, surfaces, and so on, depending on the dimension of the larger space they are in.

Subvarieties are central to questions about the structure and properties of varieties. For example, they can tell us a lot about the geometry of the variety they inhabit, including symmetry, intersections, and potential isomorphisms between different varieties. In the textbook exercise, we consider line-like subvarieties of \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) which are not present in \(\mathbb{P}^{2}\) in a comparable way, highlighting their importance in distinguishing between seemingly similar algebraic constructs.