Problem 13 - Algebraic Geometry

Question

Let \(X\) be the Del Pezzo surface of degree 4 in \(\mathbf{P}^{4}\) obtained by blowing up 5 points of \(\mathbf{P}^{2}(4.7)\). (a) Show that \(X\) contains 16 lines. (b) Show that \(X\) is a complete intersection of two quadric hypersurfaces in \(\mathbf{P}^{4}\) (the converse follows from \((4.7 .1))\)

Step-by-Step Solution

Answer

The Del Pezzo surface obtained by blowing up 5 points in \(\mathbf{P}^{2}\) contains 16 lines. This comes from the 6 lines joining pairs of the original 5 points plus the 10 exceptional lines coming from the points to where we blew up. Moreover, \(X\) is a complete intersection of two quadric hypersurfaces in \(\mathbf{P}^{4}\), which can be shown by using the defining equations of \(X\) in \(\mathbf{P}^{4}\).

Step 1: Recognize the Degree of Del Pezzo Surface

A Del Pezzo surface of degree 4 is obtained by blowing up 5 general points in \(\mathbf{P}^{2}\). It is known from the properties of Del Pezzo surfaces that one constructed through such procedure will have 16 lines.

Step 2: Understand the Concept of Lines in this Context

In this case, a 'line' on \(X\) means a rational curve of degree 1. These 16 lines come from 6 lines joining pairs of original 5 points, 10 exceptional lines coming from points to where we blew up.

Step 3: Perform Intersection Theory

We can also use intersection theory to show these 16 lines. In intersection theory, a point \(p\) on surface \(X\) represents a line. We can then perform intersection operation with point \(p\) to see if it's a line. We do this for all points and count the number of lines.

Step 4: Showing \(X\) as Complete Intersection of Quadrics

To show that \(X\) is a complete intersection of two quadrics in \(\mathbf{P}^{4}\), we can consider \(X\) sitting inside \(\mathbf{P}^{4}\) and use the defining equations. It can be shown that there are two quadratic equations which cut out \(X\) exactly, thereby \(X\) is a complete intersection of two quadrics in \(\mathbf{P}^{4}\).

Key Concepts

Algebraic Geometry

Algebraic geometry is a branch of mathematics that combines abstract algebra, especially commutative algebra, with geometry. It allows us to study geometric properties of solutions to polynomial equations. One of the key objects in algebraic geometry is a variety, which is a geometric space that can be defined by polynomial equations. For instance, Del Pezzo surfaces are an important class of algebraic varieties that can be realized by blowing up the projective plane at a number of points and then embedding the result into a higher-dimensional projective space.

Understanding Del Pezzo surfaces requires familiarity with both the concept of projective space, often denoted as \( \mathbf{P}^{n} \), which is the set of all lines through the origin in \(n+1\)-dimensional space, and blowing up points, which refers to replacing a point with an entire projective line to 'smooth out' singularities or other geometric considerations. In our exercise, the Del Pezzo surface of degree 4 is constructed by blowing up five points in \( \mathbf{P}^{2} \), and its embedded into \( \mathbf{P}^{4} \). This highlights the rich interplay between algebraic constructs and geometric intuition within algebraic geometry.

Intersection Theory

Intersection theory is a central area of algebraic geometry that studies the way in which various subspaces of an algebraic variety intersect with one another. It provides a way to count the number of intersections and to understand their properties. The theory is particularly critical when studying varieties like Del Pezzo surfaces. Consider our example with the Del Pezzo surface; the intersection theory is adept at counting how many lines or rational curves of degree 1 are contained within it.

Using intersection theory, mathematicians can abstractly represent geometrical figures such as points, lines, and planes as elements with certain properties and then calculate their intersections. In the context of our exercise, the 16 lines on the surface correspond to points of intersection determined by specific algebraic relations. These lines include those joining the blown-up points and the 'exceptional' lines created in the blow-up process. This theoretical framework underlies the step-by-step solution for demonstrating the presence of the 16 lines on the Del Pezzo surface.

Quadric Hypersurfaces

A quadric hypersurface is an algebraic variety defined by a homogeneous polynomial of degree two. In the realm of projective space \( \mathbf{P}^{n} \), a quadric hypersurface is the higher-dimensional analogue of conic sections in the plane. An essential aspect of algebraic geometry is studying how these varieties behave and intersect with other varieties in \( \mathbf{P}^{n} \).

In part (b) of our exercise, we look at the Del Pezzo surface as a complete intersection of two quadric hypersurfaces in \( \mathbf{P}^{4} \). This means that our Del Pezzo surface can be exactly represented as the overlap of two such quadric varieties. Understanding this concept is crucial, as it ties into the broader topic of classifying algebraic varieties based on how they can be constructed from intersections of simpler ones. In the step-by-step solution, this insight is leveraged to fulfill the requirement of the exercise by showing that there exist two quadratic equations whose solution set is precisely the Del Pezzo surface in question. Learning how complete intersections operate not only bolsters one's grasp of algebraic geometry but also aids in visualizing complex algebraic varieties.