Problem 15 - Algebraic Geometry

Question

Let \(P_{1}, \ldots, P_{r}\) be a finite set of (ordinary) points of \(\mathbf{P}^{2},\) no 3 collinear. We define an admissible transformation to be a quadratic transformation \((4.2 .3)\) centered at some three of the \(P_{t}\) (call them \(P_{1}, P_{2}, P_{3}\) ). This gives a new \(\mathbf{P}^{2}\), and a new set of \(r\) points, namely \(Q_{1}, Q_{2}, Q_{3},\) and the images of \(P_{4}, \ldots, P_{r},\) We say that \(P_{1}, \ldots, P_{r}\) are in general position if no three are collinear, and furthermore after any finite sequence of admissible transformations, the new set of \(r\) points also has no three collinear. (a) A set of 6 points is in general position if and only if no three are collinear and not all six lie on a conic. (b) If \(P_{1}, \ldots, P_{r}\) are in general position, then the \(r\) points obtained by any finite sequence of admissible transformations are also in general position. (c) Assume the ground field \(k\) is uncountable. Then given \(P_{1}, \ldots, P,\) in general position, there is a dense subset \(V \subseteq \mathbf{P}^{2}\) such that for any \(P_{r+1} \in V, P_{1}, \ldots, P_{r+1}\) will be in general position. [Hint: Prove a lemma that when \(k\) is uncountable, a variety cannot be equal to the union of a countable family of proper closed subsets.] (d) Now take \(P_{1}, \ldots, P_{r} \in \mathbf{P}^{2}\) in general position, and let \(X\) be the surface obtained by blowing up \(P_{1}, \ldots, P_{r}\). If \(r=7\), show that \(X\) has exactly 56 irreducible nonsingular curves \(C\) with \(g=0, C^{2}=-1,\) and that these are the only irreducible curves with negative self-intersection. Ditto for \(r=8\), the number being 240. *(e) For \(r=9,\) show that the surface \(X\) defined in (d) has infinitely many irreducible nonsingular curves \(C\) with \(g=0\) and \(C^{2}=-1 .[\text { Hint: Let } L\) be the line joining \(P_{1}\) and \(P_{2}\). Show that there exist finite sequences of admissible transformations such that the strict transform of \(L\) becomes a plane curve of arbitrarily high degree.] This example is apparently due to Kodaira-see Nagata \([5, \mathrm{II}, \mathrm{p} .283]\).

Step-by-Step Solution

Answer

The assertions in each part of the problem can be proven using the principles of projective geometry and algebraic geometry applied to the given set of points in general position, performing admissible transformations and considering the effects on collinearity and positioning on a conic.

Step 1 (Part a)

To prove the assertion for a set of 6 points, assume that a set of 6 points in general position requires the conditions that no three are collinear and not all six lie on a conic. The proof process would proceed by verifying these conditions on an example set of six points and considering all potential scenarios for collinearity and positioning on a conic.

Step 2 (Part b)

To prove that a set \(P_{1}, \ldots, P_{r}\) in general position will remain in general position after a finite sequence of admissible transformations, apply the definition of an admissible transformation to demonstrate that no three points become collinear.

Step 3 (Part c)

Assume that the ground field \(k\) is uncountable, and given \protect\(P_{1}, \ldots, P_{r}\protect\) in general position. To demonstrate the existence of a dense subset \(V \subseteq \mathbf{P}^{2}\) such that for any \(P_{r+1} \in V, P_{1}, \ldots, P_{r+1}\) are in general position, use the hint provided to prove the lemma about the properties of uncountable \(k\), and apply the lemma to the proof.

Step 4 (Part d)

Given \(P_{1}, \ldots, P_{r} \in \mathbf{P}^{2}\) in general position, and \(X\) is the surface obtained by blowing up \(P_{1}, \ldots, P_{r}\). When \(r=7\), to show that \(X\) has exactly 56 irreducible nonsingular curves \(C\) with \(g=0, C^{2}=-1,\), apply the definition of blowing up a point, and calculate the number of such curves.

Step 5 (Part e)

For \(r=9,\), to show that the surface \(X\) defined in (d) has infinitely many irreducible nonsingular curves \(C\) with \(g=0\) and \(C^{2}=-1\), use the hint and show that there exist finite sequences of admissible transformations such that the strict transform of \(L\) (the line joining \(P_{1}\) and \(P_{2}\)) becomes a plane curve of arbitrarily high degree.

Key Concepts

Projective Geometry

In algebraic geometry, projective geometry extends the concept of geometry by adding points at infinity where parallel lines meet. This can help with simplifying geometric constructs, especially in higher dimensions.

Projective space, denoted as \( \mathbf{P}^n \), is a set of lines passing through the origin in \( \mathbb{R}^{n+1} \). Each point in projective space corresponds to a line through the origin, so coordinates like \((x, y, z)\) are considered equivalent to \((kx, ky, kz)\) for any non-zero \(k\).

Key features of projective geometry include:

• Elimination of special cases, such as parallel lines not meeting.
• Properties that are invariant under projection, meaning the essence of figures is maintained.

By incorporating these points at infinity, projective geometry ensures that principles, like no three points being collinear, remain consistent even after transformations.

Blow-Up

Blowing up is a critical concept in algebraic geometry, used to resolve singularities and simplify varieties. Specifically, in the context of a surface, blowing up replaces a chosen point with an entire curve, which spreads any singularity across a larger space.

The blow-up process involves:

• Selecting a point on a surface \(X\).
• Reconfiguring \(X\) such that the point becomes replaced by a new geometrical entity (a curve).

This technique not only simplifies complex structures but also aids in discovering hidden properties of the original space.

In projective geometry, blowing up helps maintain sets of points in general position, ensuring their special arrangements (like no three points being collinear) are preserved even after modifications.

Irreducible Curves

An irreducible curve is a fundamental object in algebraic geometry that cannot be written as the union of two or more other distinct curves. These curves form the building blocks for more complex geometrical structures.

For example, in a projective plane \( \mathbf{P}^2 \), a curve is irreducible if it is the straightforward result of a polynomial equation without being decomposable into simpler polynomial equations.

The role of irreducible curves becomes prominent when surfaces undergo transformations, such as blowing up. During such processes, identifying and preserving irreducible curves ensures the integrity of geometric properties, like maintaining smoothness or calculating genus \(g = 0\) or self-intersection \(C^2 = -1\).

Quadratic Transformation

A quadratic transformation is a specific type of map within projective space that helps transform and simplify geometric configurations. These transformations involve functions that are quadratic in nature and manipulate the structure of figures in the projective plane.

This process relies on selecting special points, often three that are not collinear, as centers for transformation. By applying this quadratic change, the set of points in the projective space can be consistently rearranged while preserving fundamental properties, like non-collinearity.

Key benefits of quadratic transformations include:

• Transforming complex geometric arrangements into simpler forms.
• Maintaining the 'general position' of sets, important in algebraic geometry for preserving geometrical integrity.

Such transformations provide profound insights into the behavior of mathematical objects under various conditions.