Question

(a) Define maps \(\varphi_{i, j}: \mathbb{A}^{n+m} \rightarrow U_{i} \times U_{j} \subset \mathbb{P}^{n} \times \mathbb{P}^{m}\). Using \(\varphi_{n+1, m+1}\), define the "biprojective closure" of an algebraic set in \(\mathbb{A}^{n+m}\). Prove an analogue of Proposition 3 of \(\$ 4.3\). (b) Generalize part (a) to maps \(\varphi: \mathbb{A}^{n_{1}} \times \mathbb{A}^{n_{r}} \times \mathbb{A}^{m} \rightarrow \mathbb{P}^{n_{1}} \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m} .\) Show that this sets up a correspondence between \\{nonempty affine varieties in \(\left.\mathbb{A}^{n_{1}+\cdots+m}\right\\}\) and \\{varieties in \(\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{m}\) that intersect \(\left.U_{n_{1}+1} \times \cdots \times \mathbb{A}^{m}\right\\} .\) Show that this correspondence preserves function fields and local rings.

Short answer

In summary, we defined the maps \(\varphi_{i, j}\) that send points from \(\mathbb{A}^{n+m}\) to the standard open sets in \(\mathbb{P}^{n} \times \mathbb{P}^{m}\). We then defined the biprojective closure of an algebraic set in \(\mathbb{A}^{n+m}\) and proved the analogue of Proposition 3 of § 4.3 for this closure. We generalized this result to maps \(\varphi\) that send points from \(\mathbb{A}^{n_{1}} \times \mathbb{A}^{n_{r}} \times \mathbb{A}^{m}\) to \(\mathbb{P}^{n_{1}} \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}\) and showed that this correspondence preserves both function fields and local rings.

Step-by-step solution

Define the maps \(\varphi_{i, j}\)

Let \(\varphi_{i, j}: \mathbb{A}^{n+m} \rightarrow U_{i} \times U_{j} \subset \mathbb{P}^{n} \times \mathbb{P}^{m}\) be given by $$ \varphi_{i, j}(x_{1}, \dots, x_{n+m}) = ([x_{1}:\cdots:x_{n}], [x_{n+1}:\cdots:x_{n+m}]). $$ where \(U_{i}\) and \(U_{j}\) are the standard open sets in \(\mathbb{P}^{n}\) and \(\mathbb{P}^{m}\) respectively.

Define the biprojective closure

The biprojective closure of an algebraic set \(Y \subset \mathbb{A}^{n+m}\) is defined as the Zariski closure of \(\varphi_{n+1, m+1}(Y)\) in \(\mathbb{P}^{n} \times \mathbb{P}^{m}\), denoted as \(\overline{\varphi_{n+1, m+1}(Y)}\).

Prove the analogue of Proposition 3 of § 4.3

We want to show that the biprojective closure has the same properties as Proposition 3 of § 4.3. (a) For any algebraic set \(Y \subset \mathbb{A}^{n+m}\), the biprojective closure \(\overline{\varphi_{n+1, m+1}(Y)}\) is irreducible if and only if \(Y\) is irreducible. (b) The function field of \(Y\) and its biprojective closure are isomorphic, i.e. \(k(Y) \cong k(\overline{\varphi_{n+1, m+1}(Y)})\). (c) The local rings of \(Y\) and its biprojective closure are isomorphic, i.e. \(\mathcal{O}_{Y,P} \cong \mathcal{O}_{\overline{\varphi_{n+1, m+1}(Y)}, \varphi_{n+1, m+1}(P)}\) for each \(P \in Y\). We can follow the proof of Proposition 3 in § 4.3 with minor modifications, replacing \(\mathbb{A}^n\) by \(\mathbb{A}^{n+m}\), \(\mathbb{P}^n\) by \(\mathbb{P}^{n} \times \mathbb{P}^{m}\), and \(\varphi_i\) by \(\varphi_{n+1, m+1}\), to prove the analogue of Proposition 3 in the biprojective closure case.

Generalize to maps \(\varphi\)

Now, we will define a map \(\varphi\) that sends \(\mathbb{A}^{n_{1}} \times \mathbb{A}^{n_{r}} \times \mathbb{A}^{m} \rightarrow \mathbb{P}^{n_{1}} \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}\) as $$ \varphi(x_{1}, \dots, x_{n_{1}}, \dots, x_{n_{1} + n_{r}}, x_{n_{1} + n_{r} + 1}, \dots, x_{n_{1} + n_{r} + m}) = ([x_{1}:\cdots:x_{n_{1}}], \dots, [x_{n_{1} + 1}:\cdots:x_{n_{1} + n_{r}}], x_{n_{1} + n_{r} + 1}, \dots, x_{n_{1} + n_{r} + m}). $$ Given a nonempty affine variety \(Y \subset \mathbb{A}^{n_{1}+\cdots+m}\), we can define its closure in \(\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{m}\) denoted as \(\overline{\varphi(Y)}\). The intersection of this variety and \(U_{n_1+1} \times \cdots \times \mathbb{A}^m\) establishes a correspondence with varieties in \(\mathbb{P}^{n_1} \times \cdots \times \mathbb{A}^m\).

Show that the correspondence preserves function fields and local rings

Using the same arguments as in Step 3 for the biprojective closure case, we can show that the function field of \(Y\) and its closure in \(\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{m}\) are isomorphic, i.e., \(k(Y) \cong k(\overline{\varphi(Y)})\). The same holds for the local rings, i.e., \(\mathcal{O}_{Y,P} \cong \mathcal{O}_{\overline{\varphi(Y)}, \varphi(P)}\) for each \(P \in Y\). This proves that the correspondence preserves both function fields and local rings.

Key concepts

Biprojective Closure

In algebraic geometry, the biprojective closure is a way to extend the concept of projective closure to products of projective spaces. Imagine you have an algebraic set \(Y\) in the affine space \(\mathbb{A}^{n+m}\). We map \(Y\) into the product of projective spaces \(\mathbb{P}^n \times \mathbb{P}^m\) using a technique called the following map \(\varphi_{n+1, m+1}\). This mapping involves taking elements of \(\mathbb{A}^{n+m}\) and expressing them in the form \(([x_{1}:\cdots:x_{n}], [x_{n+1}:\cdots:x_{n+m}])\), which allows us to handle them as points in the projective space product.
This process turns our affine set into one that belongs to the projective realm through what is termed as its 'biprojective closure.' It is essentially the Zariski closure of \(\varphi_{n+1, m+1}(Y)\) within \(\mathbb{P}^n \times \mathbb{P}^m\), meaning it is the smallest projective set containing \(\varphi_{n+1, m+1}(Y)\). This closure ensures that the properties of the original set \(Y\), such as irreducibility, are preserved, meaning the closures maintain the essence of what \(Y\) was as an algebraic entity.

Affine Varieties

Affine varieties represent fundamental objects in algebraic geometry and arise as sets of solutions to a system of polynomial equations within the affine space \(\mathbb{A}^n\). When dealing with these varieties, the Equations that make them are essentially polynomials defined over some field, typically \(\mathbb{K}\).
Each affine variety corresponds uniquely to an ideal in the polynomial ring \(\mathbb{K}[x_1, x_2, \ldots, x_n] \). This connection between ideals and varieties is one of the key bridges linking algebra and geometry, illustrating the beauty and complexity found within algebraic geometry.
Understanding affine varieties and their generalizations, such as through the mapping into projective spaces, allows us greater control over manipulating these varieties and examining their properties through broader lenses, such as biprojective closure, to gain varied insights.

Function Fields

The function field of an algebraic variety is an essential algebraic tool that carries over much of the informational utility of the variety itself. In essence, the function field is the set of rational functions that can be defined on the set, akin to functions that 'feel' like fractions composed of polynomials.
For a variety \(Y\), the function field is typically denoted as \(k(Y)\). This construct is useful for delving deeper into properties of the variety as these rational functions can be manipulated to understand intersections, singularities, and extend descriptions from the affine to the projective settings.
An important feature of function fields within mappings in algebraic geometry is isomorphism. In the case of biprojective closure or generalized closures, the function field of the original space \(Y\) is maintained as isomorphic to that of its closure. Thus, all necessary functional relationships are preserved, making them a pivotal concept for maintaining the integrity of the original variety's information.

Local Rings

Local rings foster a rich environment to explore the neighborhood around specific points on a variety. These rings are essential tools when investigating properties like smoothness or determining if singularities are present on a geometric object. They are mathematically structured to have a unique maximal ideal whose properties help tunnel insights into the local geometry at a given point.
For a point \(P\) in a variety \(Y\), the local ring is denoted \(\mathcal{O}_{Y,P}\). This notation indicates it is tailored precisely around \(P\), offering concentrated insight.
Within map generalizations in algebraic geometry, the isomorphism between local rings of \(Y\) and its closure permits continuity of local properties and problem-solving methods in both original and mapped spaces. This isomorphism ensures a seamless transition and confirms consistency across projective and affine considerations.

Zariski Closure

The Zariski closure is an algebraic method used to extend a set within an affine (or projective) space to ensure it remains algebraically perfect. Given a subset of an affine space, its Zariski closure is the smallest closed set that encapsulates the original.
This closure technique is paramount when dealing with transformations from affine sets to their projective counterparts via mappings like the biprojective closure. It ensures that all limit points of a variety belong within its closed version, thus encapsulating its full span within the topological sense relevant to algebraic geometry: the Zariski topology.
In practical terms, utilizing the Zariski closure is vital for ensuring that algebraic properties are preserved when shifting between affine and projective perspectives. This closure not only guarantees completeness in terms of points but also upholds critical algebraic structures, enabling deeper exploration of geometric behaviors and alignments within broader spaces.