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Hillsdale College
Department of Mathematics and Computer Science
Honors Thesis
A Problem-Oriented Approach toElementary Algebraic Geometry
Author:
Jennifer B. Falck
Supervisor:
Dr. David Murphy
May 7, 2010
Abstract
For my mathematics honors thesis, I worked with Dr. David Murphy in order to learn some ofthe basics of Algebraic Geometry from a preprint draft of the book Algebraic geometry: A problem
solving approach [3] by the Park City Mathematics Institute 2008 Undergraduate Faculty Programbeing lead by Tom Garrity of Williams College. Dr. Murphy is one of the authors and editors of thebook. While learning from the book, we sought to proofread the book for grammatical, stylistic,and mathematical errors and to determine the effectiveness of the book as an algebraic geometrytext.
1
Acknowledgments
I would like to thank to Dr. David Murphy for his diligent work throughout the year. As an authorand editor of Algebraic Geometry: A Problem Solving Approach [3], he shared in both my successesand my frustrations. His personal investment and organization encouraged my progress throughthe material over the course of the year. His dedication, patience, and comprehensive nature aidedmy overall understanding of elementary algebraic geometry.
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Contents
1 Introduction 4
2 Introduction to Algebraic Geometry 4
3 Conics 5
3.1 Conics in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Conics in C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Conics in P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Parabolas, Ellipses, and Hyperbolas as Spheres . . . . . . . . . . . . . . . . . . . . . 103.5 Degenerate Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 Singular and Smooth Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Cubics 13
4.1 Cubics as a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Inflection Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Hessians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Normal Forms of Cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Higher Degree Curves 16
5.1 Genus of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Bezout’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Elementary Affine Geometry 19
6.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Hilbert Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 The Coordinate Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4 Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 An Introduction to Algebraic Geometry: A Problem Solving Approach 22
7.1 Experience Editing and Revising the Book . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Comments on the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8 Conclusion 24
3
1 Introduction
A traditional undergraduate mathematics major curriculum consists of courses in calculus, geome-try, algebra, and analysis. These traditional undergraduate majors rarely have the opportunity totake courses in subjects like algebraic geometry, which have been exclusively reserved for graduatestudents until recently. However, within the past twenty years a number of texts on algebraic geom-etry have been written with the intent to introduce this important topic to the larger undergraduateaudience.
One of the most popular of these new books, Ideals, Varieties, and Algorithms: An Introduction
to Computational Algebraic Geometry and Commutative Algebra [2], seeks to introduce algebraicgeometry to undergraduate students using a computational approach. The book written by the ParkCity Mathematics Institute 2008 Undergraduate Faculty Program [3] aims to teach the classicalmaterial, which includes curves, using a contemporary, problem-oriented method.
I will give an introduction to algebraic geometry following the presentation in Algebraic Ge-
ometry: a Problem Solving Approach [3]. Our focus will be on algebraic curves. Specifically, wewill explore conics, cubics, and higher degree curves. After we have explored these curves, we willexamine elementary affine geometry. After this brief introduction to algebraic geometry, I willdiscuss my experience of editing and revising the book and offer some evaluative comments on thebook.
2 Introduction to Algebraic Geometry
Before we can explore specific topics within algebraic geometry, we must understand the overlyingstructure and purpose of the field. Algebraic geometry primarily seeks to create a link betweenalgebra and geometry. It combines algebraic techniques and properties with geometric objects inorder to portray a more complete understading of both the algebra and the geometry of an object.
We will begin with a simple example to illustrate the connection between algebra and geometry.
Example 1. We have two corresponding representations.
1. The algebraic representation is the polynomial x2 + y2 − 1 = 0.
2. The geometric representation is the unit circle in R2.
Throughout our study of the fundamentals of algebraic geometry, we will continually come back tothis idea of representing one idea both algebraically and geometrically.
The algebraic geometry, which we will explore, can be divided into two primary groups: classicalalgebraic geometry of curves and elementary affine geometry. Classical algebraic geometry wasestablished in the time ranging from Descartes to Riemann. This classical algebraic geometryconsists of the study of algebraic curves and also extends to surfaces. Within classical algebraicgeometry, our focus will be on algebraic curves. The other section, elementary affine geometry, willfocus on applying resources from abstract algebra and topology to describe geometric curves andsurfaces. The elementary affine geometry, which we will explore, was primarily established in the1930s to 1950s. By presenting these connections between algebra and geometry, we will constructa basis for elementary algebraic geometry.
Since our primary focus will be curves, we will begin our study of algebraic geometry by intro-ducing algebraic curves.
Definition 1. A curve, denoted C or V (P ), is the set
{(x, y) : P (x, y) = 0},
4
where P (x, y) is a polynomial.
Using this notation, we will refer to V (P ) as the vanishing of the polynomial P or the zero set ofP . To clarify the idea of an algebraic curve, we will look at a brief example.
Example 2. Let P be the polynomial x2 + y2 − 1. Then V (P ) = {(x, y) : x2 + y2 = 1}.
This example shows that an algebraic curve is the set of the points in the zero set of the polynomial.In this particular case, the algebraic curve geometrically can be represented as the set of all thepoints on the unit circle.
With this summary of the goals and design of algebraic geometry, we will begin our investigationof the subject’s finer details.
3 Conics
Our first set of curves that we will discuss are the conics. As with all curves, we are interestedin describing them as the zero sets of polynomials. Specifically, conics are the zero sets of seconddegree polynomials. To simplify these conics, we will categorize them into three classes: parabolas,ellipses, and hyperbolas.
Each of these three classes are commonly represented algebraically using the following zero sets.
1. A parabola often is typified by the set {(x, y) : y = x2}.
2. An ellipse often is typified by the set {(x, y) : x2 + y2 = 1}.
3. A hyperbola often is typified by the set {(x, y) : x2 − y2 = 1}.
One of our tools to analyze these classes of conics is a change of coordinates. A change ofcoordinates can be described as a dictionary between coordinate systems. We are creating aninvertible mapping between two coordinate systems so that we may examine the properties of thecurve that are invariant under a change of coordinates. One of these invariants we will consider isthe equivalence between two conics under a change of coordinates.
Definition 2. The zero loci of two conics are equivalent under a change of coordinates if thedefining polynomial for one of the conics can be transformed via a change of coordinates into thedefining polynomial of the other conic.
We will use this equivalence of two conics under a change of coordinates to prove that parabolas,ellipses, and hyperbolas are one class instead of three distinct classes of conics.
3.1 Conics in R2
Suppose we require the elements of the zero set of a second degree polynomial to be in R2. With
this requirement, we gain information about our conics. We also gain information to give a morerigorous definition of a change of coordinates in this setting.
Definition 3. A real affine change of coordinates in R2 is given by the system of equations
u = ax + by + e
v = cx + dy + f,
where a, b, c, d, e, f ∈ R and ad − bc 6= 0.
5
We can visualize a real affine change of coordinates in terms of a change in the orientation ofeach axis with respect to the conic. A real affine change of coordinates may stretch, compress, orshift the axes. It may rotate or reflect them. We may have a real affine change of coordinates whichpreserves the angle between the axes or we may have a real affine change of coordinates in whichthe angle between the axes is not preserved.
Example 3. Consider the change of coordinates
u = x − 3,
v = y − 2.
The unit circle in the (u, v)-coordinate system, which is represented using the equation
u2 + v2 − 1 = 0,
is expressed in the (x, y)-coordinate system by the equation
(x − 3)2 + (y − 2)2 − 1 = x2 − 6x + y2 − 4y + 12 = 0.
1 2−1−2
1
2
−1
−2
u
v
0 1 2 3 40
1
2
3
4
x
y
In this example, the real affine change of coordinates is merely a shift of each axis while preservingthe angle between the axes.
Example 4. Consider the change of coordinates
u = x,
v = x + y.
The unit circle in the (u, v)-coordinate system, which is represented using the equation
u2 + v2 − 1 = 0
is expressed in the (x, y)-coordinate system by the equation
x2 + (x + y)2 − 1 = 2x2 + 2xy + y2 − 1 = 0.
1 2−1−2
1
2
−1
−2
u
v
1 2−1−2
1
2
−1
−2
x
y
6
In this example, the real affine change of coordinates does not preserve the angle between the axes,which is reflected in the distortion of the circle in the (x, y)-coordinate system. Example 3 andExample 4 provide us with evidence of a few of the transitions that can occur by using a real affinechange of coordinates.
We will now consider the equivalences that occur under a real affine change of coordinates.Geometrically the capabilities of a real affine change of coordinates include stretching, compressing,shifting, rotating, and reflecting within the real plane. Consider an arbitrary ellipse. First we canshift the axes so that the origin is in the middle of the ellipse. Second we can rotate the axes suchthat the major axis of the ellipse is along one axis of the coordinate system and the minor axis ofthe ellipse is along the other axis of the coordinate system. Third we can contract or stretch theaxes such that the ellipse intersects each axis at 1. Using these steps, any ellipse can be transformedinto the unit circle under the real affine change of coordinates. Therefore, all ellipses are equivalentunder an affine change of coordinates. The argument is similar for parabolas and hyperbolas aswell.
Theorem 1. All ellipses in R2 are equivalent, all parabolas in R
2 are equivalent, and all hyperbolas
in R2 are equivalent under a real affine change of coordinates.
This is the first theorem leading to our proof that all parabolas, ellipses, and hyperbolas areequivalent under a change of coordinates. Unfortunately, there is no real affine change of coordinatesthat maps an ellipse to a hyperbola, a parabola to an ellipse, or a parabola to a hyperbola. The threeclasses of conics are distinct under a real affine change of coordinates. When we think about thisgeometrically, we are faced with a few facts. Parabolas and ellipses have one continuous connectedpart, but hyperbolas have two. Ellipses are bounded along both axes, but parabolas are not. Withthese facts in mind, we notice that the capabilities of real affine change of coordinates do not allowcutting or gluing. Therefore, we need more freedom in our tools to establish an equivalence betweenthe three classes of conics.
3.2 Conics in C2
If we require the solutions in the zero set of a second degree polynomial to be elements in R2, then
some zero sets will be empty.
Example 5. Suppose we have P (x, y) = x2 + 1.In the real plane, V (P ) = {(x, y) ∈ R
2 : x2 = −1} = ∅.In the complex plane, V (P ) = {(x, y) ∈ C
2 : x2 = −1} = {(±i, y) : y ∈ C}.
By working in the complex plane, C2, the zero set of a second degree polynomial will never be
empty.Suppose we require the elements of the zero set of a second degree polynomial be in C
2. With thisexpansion to include imaginary numbers, we will need to define a different change of coordinates.
Definition 4. A complex affine change of coordinates in C2 is given by the system of equations
u = ax + by + e
v = cx + dy + f,
where a, b, c, d, e, f ∈ C and ad − bc 6= 0.
Using this newly defined change of coordinates, we want to examine equivalence under a complexaffine change of coordinates.
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Example 6. Consider the complex affine change of coordinates
u = x,
v = iy.
The unit circle in the (u, v)-coordinate system, which is given by the equation
u2 + v2 − 1 = 0,
is expressed in the (x, y)-coordinate system by the equation
x2 + (iy)2 − 1 = x2 − y2 − 1 = 0,
which is a hyperbola geometrically.
Using Example 6 and Theorem 1, we deduce the next theorem.
Theorem 2. Under a complex affine change of coordinates, all ellipses and hyperbolas are equiva-
lent.
However, parabolas are still a distinct class under a complex affine change of coordinates.Therefore, we are still looking for a change of coordinates with all the capabilities necessary todeclare parabolas, ellipses, and hyperbolas equivalent.
3.3 Conics in P2
We have established the equivalence of ellipses and hyperbolas under a complex affine change ofcoordinates, but parabolas remain a distinct class under a complex affine change of coordinates. Toremedy this distinction, we will work in the complex projective plane, which introduces a differentunderstanding of the points at infinity in C.
Before we can establish the complex projective plane, we must describe its points. We beganwith points in R
2, which are ordered pairs of real numbers. This understanding of points is gen-eralized to apply to the complex plane. Thus, ordered pairs of complex numbers are consideredpoints in C
2. The transition into points in the complex projective plane is less natural but allowsus to treat points at infinity of C
2 in a mathematically rigorous way.
Definition 5. If (x, y, z) and (u, v, w) are points in C3−{(0, 0, 0)}, we say that (x, y, z) is equivalent
to the point (u, v, w) if and only if (x, y, z) = (λu, λv, λw) for some λ ∈ C − {0}. For a point(x, y, z) ∈ C
2 − {(0, 0, 0)}, its equivalence class is
(x : y : z) = {(u, v, w) ∈ C3 − {(0, 0, 0)} : (u, v, w) = (λx, λy, λz) for some λ ∈ C − {0}}.
With this new appreciation for equivalent points, we can define the complex projective plane.
Definition 6. The complex projective plane, P2(C) or P
2, is the set {(x : y : z) for all points(x, y, z) ∈ C
3 − {(0, 0, 0)}}.
The complex projective plane can be difficult to visualize, but we can think of P2 as the union of C
2
and the line at infinity, where the line at infinity is the set of points {(x : y : z) ∈ P2(C) : z = 0}.
With this new concept of the complex projective plane, restrictions need to be placed upon thepolynomials defining the zero sets in P
2.
8
Definition 7. A polynomial is homogeneous if the sum of the exponents in every monomial is thesame. This common sum is called the homogeneous degree of the polynomial.
To illustrate this definition, we will look at a few examples of homogeneous and non-homogeneouspolynomials.
Example 7.
1. The polynomials x2 + y2 − z2 and xz − y2 are homogeneous of homogeneous degree 2.
2. The polynomials x2 + y2 − 1 and x − y2 are non-homogeneous.
If P (x, y, z) is a homogeneous polynomial of homogeneous degree m, then P (λx, λy, λz) =λmP (x, y, z). Thus the zero sets of homogeneous polynomials are well-defined in P
2. However, thezero sets of non-homogenous polynomials are not well-defined in P
2. Thus, we should only describea zero set of homogeneous polynomial in P
2.Not all zero sets in C
2 are of homogeneous polynomials. In order to work with them in P2, we
must have a way to homogenize these non-homogeneous polynomials. We will homogenize non-homogeneous polynomials by using the following procedure. Identify the monomial of the highestdegree n. For a monomial of a degree m not equal to n, we will multiply that monomial by zn−m.By doing this to every monomial of degree not equal to n, we homogenize the polynomial.
Example 8. If P (x, y) = x2 + y2 − 1, then the homogenized polynomial is x2 + y2 − z2.
Just as we needed a way to homogenize a polynomial, we also need a way to reverse the homog-enization or dehomogenize the polynomial. To dehomogenize a homogeneous polynomial, we setz = 1.
Example 9. Suppose we have the homogeneous polynomial P (x, y, z) = x2 + y2 − z2. ThenP (x, y, 1) = x2 + y2 − 12 = x2 + y2 − 1.
Now that we have determined the process to produce homogeneous polynomials, we are able toexamine the zero sets of these polynomials in P
2.Just as we had a change of coordinates as a tool in the real plane and in the complex plane, we
will define a change of coordinates specific to the projective plane.
Definition 8. A projective change of coordinates is given by the system of equations
u = a11x + a12x + a13x
v = a21x + a22x + a23x
w = a31x + a32x + a33x,
where aij ∈ C and
det
a11 a12 a13
a21 a22 a23
a31 a32 a33
6= 0.
Using projective changes of coordinates, we establish parabolas, ellipses, and hyperbolas in P2 as a
single class.
Theorem 3. All parabolas, ellipses, and hyperbolas are equivalent under a projective change of
coordinates.
9
We illustrate this result by showing that the parabola v−u2 = 0 in the (u, v)-coordinate systemis equivalent under a projective change of coordinates to the unit circle x2 + y2 − 1 = 0 in the(x, y)-coordinate system.
Example 10. Consider the parabola v−u2 = 0 in the (u, v)-coordinate system for C2. Homogenize
to obtain the corresponding parabola vw − u2 = 0 in P2. Then, by the projective change of
coordinates
u = z
v = x + iy
w = x − iy
the parabola becomes (x + iy)(x − iy) − z2 = x2 + y2 − z2 = 0 in the (x : y : z)-coordinate systemfor P
2. Upon dehomogenization we obtain the equation x2 + y2 − 1 = 0, which is the unit circle inthe (x, y)-coordiniate system for C
2.
3.4 Parabolas, Ellipses, and Hyperbolas as Spheres
Now that we have established the equivalence between parabolas, ellipses, and hyperbolas in P2,
we can talk about some of the properties that they share as a single class of conics. One ofthese properties is their topological equivalence to Euclidean spheres. In order to determine thisequivalence, we need to understand the complex projective line.
With the understanding of the complex projective plane, we can similarly describe and definethe complex projective line. We need to redefine the equivalence of points, just as we did for theprojective plane.
Definition 9. If (x, y) and (u, v) are points in C2 −{(0, 0)}, we say that (x, y) is equivalent to the
point (u, v) if and only if (x, y) = (λu, λv) for λ ∈ C − {0}. If (x, y) ∈ C2 − {(0, 0)}, denote its
equivalence class by
(x : y) = {(u, v) ∈ C2 − {(0, 0)} : (u, v) = (λx, λy) for λ ∈ C − {0}}.
We will define the complex projective line with this redefinition of equivalent points.
Definition 10. The complex projective line, P1, is the set {(x : y) : (x, y) ∈ C
2 − {(0, 0)}}.
Just as the complex projective plane is difficult to visualize, the complex projective line presentsthe same difficulty. We can picture P
1 as the union of C and a single point at infinity, where thepoint at infinity is the point (1 : 0).
The complex projective line P1 is topologically equivalent to a sphere, since we can construct a
continuous map with a continuous inverse between P1 and a standard Euclidean sphere in R
3. Wecan also find a bijective polynomial map from P
1 to any ellipse, hyperbola, or parabola.
Theorem 4. All ellipses, hyperbolas, and parabolas are topologically equivalent to standard Eu-
clidean spheres.
To complement the algebraic representation of the equivalence of parabolas, ellipses, and hyperbolasin P
2 given by the projective change of coordinates, this theorem provides us with a geometric wayto understand the equivalence between the three conics.
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3.5 Degenerate Conics
We will extend our study of conics from parabolas, ellipses, and hyperbolas to include the degenerateconics. The degenerate conics are crossing lines and double lines.
Definition 11. Crossing lines consist of the zero set of a polynomial of the form (a1x + b1y +c1z)(a2x + b2y + c2z), where at least one of
det
[
a1 b1
a2 b2
]
, det
[
a1 c1
a2 c2
]
, or det
[
b1 c1
b2 c2
]
is nonzero.
If the first determinant is nonzero, the zero set consist of two lines that intersect at a point wherez 6= 0. If the first determinant is 0 but one of the others is not, the lines intersect at a point wherez = 0.
Example 11.
1. The zero set of the polynomial P (x, y, z) = (x)(y) is a crossing lines conic.
2. The zero set of the polynomial P (x, y, z) = (−x+ y + z)(2x+ y +3z) is a crossing lines conic.
The set of crossing lines is a distinct class of conics in P2 under a projective change of coordinates. It
is algebraically distinguished from parabolas, ellipses, and hyperbolas in that its defining equationis reducible. A further degeneration of crossing lines allows all three of the determinants to be 0.In this case, the polynomial is a scalar multiple of a perfect square of a linear polynomial.
Definition 12. Double lines consist of the the zero set of a polynomial of the form (a1x+b1y+c1z)2.
Example 12.
1. The zero set of the polynomial P (x, y, z) = x2 is a double lines conic.
2. The zero set of the polynomial P (x, y, z) = (−x + y + z)2 is a double lines conic.
The set of double lines is a distinct class of conics in P2 under a projective change of coordinates.
Therefore there are three distinct classes of conics in P2 under a projective change of coordinates:
1. Parabolas, ellipses, and hyperbolas,
2. Crossing lines, and
3. Double lines.
3.6 Singular and Smooth Points
We have explored properties that unify the classes of conics in P2. Now we will examine a difference
between the classes of conics in P2. This difference can be found in the number of singular points
in each class.
Definition 13. A point p = (a : b : c) is singular if it is a point on a curve C = {(x : y : z) ∈ P2 :
f(x, y, z) = 0}, where f(x, y, z) is a homogeneous polynomial, and
∂f
∂x(a, b, c) = 0,
∂f
∂y(a, b, c) = 0,
∂f
∂z(a, b, c) = 0.
A point that is not singular is smooth.
11
Basically, a point is singluar if the partial derivatives with respect to each of the variables of thehomogeneous polynomial generating the curve equals zero, which means the curve does not have atangent line at the point.
Example 13. Let P (x, y, z) = (x + y − z)(x + y + z). Then we have the following:
∂f
∂x(1,−1, 0) = (1 − 1 − 0) + (1 − 1 + 0) = 0,
∂f
∂y(1,−1, 0) = (1 − 1 − 0) + (1 − 1 + 0) = 0,
∂f
∂z(1,−1, 0) = (1 − 1 − 0) − (1 − 1 + 0) = 0.
Therefore the point (1,−1, 0) is a singular point on the curve V (P ).
Example 14. Let P (x, y, z) = (x + y − z)(x + y + z). Then we have the following:
∂f
∂x(1, 0, 1) = (1 + 0 − 1) + (1 + 0 + 1) = 2 6= 0,
∂f
∂y(1, 0, 1) = (1 + 0 − 1) + (1 + 0 + 1) = 2 6= 0,
∂f
∂z(1, 0, 1) = (1 + 0 − 1) − (1 + 0 + 1) = −2.
Therefore the point (1, 0, 1) is a smooth point on the curve V (P ).
We can use the classification of singular and smooth points to create a classification betweensingular and smooth curves.
Definition 14. If a curve has at least one singular point, the curve is singular. If a curve has nosingular points, the curve is smooth.
Example 15.
1. Let f(x, y, z) = (x + y − z)(x + y + z). Since (1,−1, 0) is a singular point on the curve V (f)by Example 12, V (f) is a singular curve.
2. Let f(x, y, z) = x2 + y2 − z2. Then we have the following:
∂f
∂x(a, b, c) = 2a,
∂f
∂y(a, b, c) = 2b,
∂f
∂z(a, b, c) = −2c.
The only point that would allow all three partial derivatives to be zero is (0 : 0 : 0). However,(0 : 0 : 0) /∈ P
2 by the definition of complex projective plane. Therefore V (f) is a smoothcurve.
Theorem 5. All parabolas, ellipses, and hyperbolas are smooth curves. Crossing lines have exactly
one singular point. All points in double lines are singular.
12
As we proceeded through our study of conics, we made a transition from the real plane tothe complex plane, and from the complex plane to the complex projective plane. We did thisin order to ease our algebraic calculations at the expense of our ability to visualize the geometricrepresentations. To acheive the goal of declaring parabolas, ellipses, and hyperbolas to be equivalentunder a change of coordinates, we need to work in the complex projective plane P
2. However, ifour goal was to create ease in the visualization of a curve, we would use the real plane R
2. As weexpanded our defintion of plane, we analyzed some of the properties within classes of conics andbetween classes of conics in the complex projective plane.
4 Cubics
We have discussed the classes of conics under a projective change of coordinates and presentedsome properties of these classes. We will now move on to cubics. Just as the conics are zero setsof second degree polynomials, cubics are the zero sets of third degree polynomials. We will workin C
2 for the zero sets of polynomials in two variables. Whereas, we will work in P2 for the zero
sets of homogeneous polynomials in three variables.
4.1 Cubics as a Group
By identifying smooth cubics as a group, we can apply the techniques and properties found in grouptheory to the zero sets of third degree polynomials. In order to identify smooth cubics as a group,we want to recall the definition of a group from abstract algebra.
Definition 15. Let G be a set of elements equipped with the binary operation (∗). We call G agroup if, for all g1, g2, g3 ∈ G:
1. The binary operation is associative such that
(g1 ∗ g2) ∗ g3 = g1 ∗ (g2 ∗ g3).
2. There exists an (unique) identity element e ∈ G such that
g1 ∗ e = e ∗ g1 = g1.
3. For each g ∈ G, there exists an (unique) inverse element h ∈ G such that
g ∗ h = h ∗ g = e.
Smooth cubic curves are commutative groups equipped with the commutative binary operation+. In P
2(C), any line intersects a smooth cubic in three points counting multiplicities. Thus, if Pand Q are two points on a smooth cubic C, the line through P and Q intersects C in a unique thirdpoint, PQ. This is a binary operation on the smooth cubic, called the chord-tangent compositionlaw. This operation does not define a group structure on C but is used to create one. Let + bedefined relative to a specific point of inflection O on the curve. For points P, Q on the curve, P +Qis the intersection of the line through PQ and O with the curve.
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4.2 Inflection Points
In order to comprehend the function of the binary operation + with which smooth cubics areequipped, we must understand the definition of + relative to a point of inflection. To accomplishthis, we must define a point of inflection.
Definition 16. Let P (x, y) be a homogeneous polynomial. A root is a point (a : b) where P (a, b) =0. A root has multiplicity k if (bx− ay)k divides P (x, y) and (bx− ay)k+1 does not divide P (x, y).
The definition of the intersection multiplicity of a curve and a line builds on the concept of themultiplicity of a root.
Definition 17. The intersection multiplicity of a curve V (P ) and a line V (l) in P2 at the point
(x0 : y0 : z0) is the multiplicity of the root (x0 : z0) of P (x, ax + cz, z).
Using the the definition of the intersection multiplicity of a curve and a line, we can construct thedefinition of an inflection point.
Definition 18. Let P (x, y, z) be an irreducible, degree n, homogeneous polynomial. An inflection
point is a non-singular point p in V (P ) ⊂ P2 such that the tangent line to the curve at the point p
and V (P ) have intersection multiplicity of at least 3.
Example 16. Let P (x, y, z) = x3 − y2z + z3.
1. The tangent line to V (P ) at the point (2 : 3 : 1) has intersection multiplicity 2. Thereforethe point (2 : 3 : 1) is not an inflection point in V (P ).
2. The tangent line to V (P ) at the non-singular point (0 : 1 : 1) has intersection multiplicity 3.Therefore the point (0 : 1 : 1) is an inflection point in V (P ).
With the available techniques, one must test each point on the curve to identify an inflectionpoint. To identify an inflection point, the current process includes the steps:
1. Select a non-singular point p = (a : b : c) on a curve V (P ), where P (x, y, z) is an irreducible,degree n, homogeneous polynomial.
2. Find the tangent line to a curve at the point p.
3. Calculate P (x, ax + cz, z).
4. Rewrite P (x, ax+ cz, z) such that P (x, ax+ cz, z) = (cx+az)kf(x, y, z) where (cx+az) doesnot divide f(x, y, z).
5. The intersection multiplicity of V (P ) and the tangent line at p is k.
6. If k ≥ 3, then p is an inflection point of the curve V (P ).
The commutative binary operation + on the group of smooth cubics is defined relative to aninflection point on the curve. The number of different inflection points by which we can define +on smooth cubic curves is an invariant.
Theorem 6. Every smooth cubic curve has exactly nine points of inflection counting multiplicities.
Therefore, each smooth cubic curve can be equipped with nine possible forms of the commutativebinary operation + to produce nine possible commutative groups.
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4.3 Hessians
Our current process of identifying inflection points can be a tedious process, since we need toindividually check each point on a curve. Hessians aid us in the discovery of a more efficient wayto recognize these inflection points.
Definition 19. Let P (x, y, z) be a degree n, homogeneous polynomial. Its Hessian, denoted H(P ),is
H(P )(x, y, z) = det
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz
,
where
Pxx =∂2P
∂x2, Pxy =
∂2P
∂x∂y, and so forth.
The curve V (H(P )) is called the Hessian curve.
Theorem 7. The curve V (H(P )) is invariant under a projective change of coordinates.
Theorem 8. Let P (x, y, z) be a degree n, homogeneous polynomial. If the curve V (P ) is smooth,
then p ∈ V (P ) ∩ V (H(P )) if and only if the point p is an inflection point of V (P ).
Therefore, rather than testing each individual point in V (P ) to identify the inflection points, wecan focus on the points in the intersection V (P ) ∩ V (H(P )).
4.4 Normal Forms of Cubics
Now we will look at two normal forms of cubics: the Weierstrass normal form and the canonicalform. These normal forms state that every smooth cubic can be written in a particular form. Theseforms are useful and provide a nice way to characterize smooth cubics.
Definition 20. Every smooth cubic is projectively equivalent to a cubic of the form y2 = x3 +Ax + B, where A and B are uniquely determined for each cubic. This is called the Weierstrass
normal form.
Example 17. Consider the equation y2 = x3 + 6x2 − 2x + 5. Using the change of coordinates
u = x + 2
v = y,
we get the equation
v2 = (u − 2)3 + 6(u − 2)2 − 2(u − 2) + 5
= (u3 − 6u2 + 12u − 8) + (6u2 − 24u + 24) − 2(u − 2) + 5
= u3 − 10u + 25.
Therefore, we transformed the equation y2 = x3 + 6x2 − 2x + 5 into the Weierstrass normal formby a change of coordinates.
The other normal form, which we will analyze is the canonical form.
Definition 21. Every smooth cubic is projectively equivalent to a cubic of the form y2 = x(x −1)(x − λ). This is called the canonical form.
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Example 18. Consider the equation y2 = 3x3 +9x2
2+
3x
2. Using the change of coordinates
u = 3x
v = y,
we get the equation
v2 = u3 +3u2
2+
u
2
= u(u2 −3u
2+
1
2)
= u(u − 1)(u −1
2).
For a smooth cubic written in canonical y2 = (x − 1)(x − λ), the value of λ is not uniquelydetermined. There are six possible choices for λ, which are the following values:
λ,1
λ, 1 − λ,
1
1 − λ,
λ − 1
λ,
λ
λ − 1.
Using any of these six choices, the canonical form will yield the same smooth cubic.
5 Higher Degree Curves
We have discussed conics, which are the zero sets of second degree polynomials. We have discussedcubics, which are the zero sets of third degree polynomials. We want to be able to extend ourstudy of zero sets of polynomials to higher degree polynomials or higher degree curves. Higherorder curves are the zero sets of higher degree, homogeneous polynomials in P
2. When examininga higher order curve, it is often beneficial to factor the polynomial into irreducible polynomials andanalyze the properties of these parts.
We will assume that all polynomials in Section 5 are irreducible unless otherwise stated.
5.1 Genus of a Curve
Just as we established the topological equivalence of the smooth conics with the standard Eu-clidean sphere in Section 3.4, we would like to establish topological equivalences with higher degreecurves. We can establish a geometric and an algebraic method of establishing invariants to aid indetermining the topological equivalence of curves.
Definition 22. Let V (P ) be a smooth, irreducible curve in P2. The topological genus of the curve
V (P ) is the number of holes in the corresponding real surface.
The topological genus of a curve is a geometric invariant of V (P ). We can use the topologicalgenus to determine the topological equivalence of smooth curves.
Definition 23. Let V (P ) be a curve of degree d. Then the arithmetic genus of the curve V (P ) is
(d − 1)(d − 2)
2.
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The arithmetic genus of a curve is an algebraic invariant of V (P ). We can use the arithmeticgenus to determine the equivalence of smooth curves algebraically.
From Section 3.4, we know that every smooth conic curve is topologically equivalent to a realEuclidean sphere, which has no holes or has topological genus 0. If we compute the arithmetic
genus of these second degree curves, we find that the arithmetic genus is(1)(0)
2= 0, which is equal
to the topological genus.Every smooth cubic curve is topologically equivalent to a real torus, which has one hole or has
topological genus 1. If we compute the arithmetic genus of these third degree curves, we find that
the arithmetic genus is(2)(1)
2= 1, which is equal to the topological genus.
Every smooth quartic curve is topologically equivalent to a sphere with three holes or hastopological genus 3. If we compute the arithmetic genus of these fourth degree curves, we find that
the arithmetic genus is(3)(2)
2= 3, which is equal to the topological genus.
Every smooth quintic curve is topologically equivalent to a sphere with six holes or has topo-logical genus 6. If we compute the arithmetic genus of these fifth degree curves, we find that the
arithmetic genus is(4)(3)
2= 6, which is equal to the topological genus.
Therefore, we see that we can determine the equivalence of smooth curves both algebraicallyand geometrically without contradiction with the use of the genus.
5.2 Bezout’s Theorem
In Section 3, we moved into planes in which algebraic calculations were simpler, but geometricvisualizations became more difficult. When we discussed the genus, we looked at both algebraicand geometric ways to categorize smooth curves. Bezout’s Theorem continues to promote thisrelationship between the algebraic and geometric. The theorem states an equivalence betweensomething geometric (the intersection multiplicity of two curves) and something algebraic (theproduct of the degrees of polynomials). Now we want to further exploit algebraic techniques byusing abstract algebra.
As we extend our study of curves to higher degrees, we would like to study theorems that can beapplied to homogeneous curves of any degree. Bezout’s Theorem is a critical theorem in algebraicgeometry that can be applied to homogeneous curves of any degree.
We will explore some of the motivating factors of Bezout’s Theorem. Recall the definition ofthe multiplicity of a root for a polynomial in C[x] and the Fundamental Theorem of Algebra.
Definition 24. Let f(x) be a polynomial in C[x]. The multiplicity of the root a of f(x) is m iff(x) = (x − a)mg(x) for m > 0 such that (x − a) does not divide g(x).
Theorem 9 (Fundamental Theorem of Algebra). If f(x) is a degree d polynomial in C[x], then
f(x) = (x − a1)m1(x − a2)
m2 · · · (x − an)mn ,
where∑n
i=1 mi = d and ai is a complex root of f(x) with multiplicity mi.
The Fundamental Theorem of Algebra discusses the multiplicity of complex roots. This leadsus to consider a more generalized definition of multiplicity. By making this generalization, wecan consider the expansion of the Fundamental Theorem of Algebra to a statement about theintersections of curves.
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Definition 25. Let f be a non-homogeneous polynomial with point p in the set V (f). Themultiplicity of f at p, denoted mpf , is the degree of the lowest degree non-zero term in the Taylorseries expansion of f at p.
We want a theorem regarding the intersection of two curves defined by polynomials f and g.We would like to be able to use an application of the Fundamental Theorem of Algebra to describethe intersection of two curves.
Theorem 10 (Bezout’s Theorem). Let f and g be homogeneous polynomials in C[x, y, z] with no
common component. Let V (f) and V (g) be the corresponding curves in P2. Then
∑
p∈V (f)∩V (g)
I(p, V (f) ∩ V (g)) = (deg f)(deg g),
where I(p, V (f) ∩ V (g)) is the intersection multiplicity of p and V (f) ∩ V (g).
The primary tool in the proof of Bezout’s Theorem is the resultant. The resultant of twopolynomials is used to find the common roots of the curves.
Definition 26. The resultant of two polynomials f(x) = anxn + an−1xn−1 + · · · + a1x + a0 and
g(x) = bmxm+bm−1xm−1+· · ·+b1x+b0, denoted Res(f, g), is the determinant of the (m+n)×(m+n)
matrix
an an−1 · · · a0 0 0 · · · 00 an an−1 · · · a0 0 · · · 0
0 0. . .
. . . · · · · · · · · · · · ·0 0 · · · 0 an an−1 · · · a0
bm bm−1 · · · b0 0 0 · · · 00 bm bm−1 · · · b0 0 · · · 0
0 0. . .
. . . · · · · · · · · · · · ·0 0 · · · 0 bm bm−1 · · · b0
,
where there are m rows containing some ai and n rows containing some bi.
Example 19.
1. Suppose f(x) = x2 + 2x + 3 and g(x) = 4x − 1. Then Res(f, g) is the determinant of thematrix
1 2 34 −1 00 4 −1
.
The determinant of this matrix is (1) + (0) + (48) − (0) − (0) − (−8) = 57 6= 0.
2. Suppose f(x) = x3 + x and g(x) = x. Then Res(f, g) is the determinant of the matrix
1 0 1 01 0 0 00 1 0 00 0 1 0
.
The determinant of this matrix is 0.
The two polynomials f(x) = anxn + an−1xn−1 + · · ·+ a1x+ a0 and g(x) = bmxm + bm−1x
m−1 +· · ·+ b1x+ b0 have a common root when Res(f, g) = 0. Therefore the polynomials in Example 18.1do not have a common root, and the polynomials in Example 18.2 have a common root.
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6 Elementary Affine Geometry
Thus far we have studied algebraic curves and results from classical algebraic geometry. Now wewill present a treatment of algebraic geometry based on abstract algebra developed by Zariski inthe first half of the twentieth century. In our transition, we will focus on ring theory to describegeometric features.
In elementary affine geometry, we will work in the affine n-space over a field k.
Definition 27. An affine n-space over a field k is the set An(k) = {(a1, a2, . . . , an) : ai ∈ k for
i = 1, . . . , n}.
Notice that A2(R) is R
2 under this notation, and A1(C) is C. By using A
n(k), we can use anyfield k, which means that the coefficients of the polynomial are not constrained by R or C.
Just as with curves, in affine geometry we are interested in the zero sets of polynomials. Wecall these zero sets algebraic sets.
Definition 28. The algebraic set defined by a set of polynomials S contained in k[x1, x2, . . . , xn]is the set V (S) = {(a1, a2, . . . , an) ∈ A
n(k) : P (a1, a2, . . . , an) = 0 for all P ∈ S}.
Definition 29. An affine variety is an irreducible algebraic set.
Example 20.
1. The empty set is an algebraic set in An(k), where ∅ = V (1).
2. The set An(k) is an algebraic set in A
n(k), where An(k) = V (0)
3. If X and W are algebraic sets in An(k), then X ∪ W and X ∩ W are also algebraic sets in
An(k).
Ideals are a tool used in the proof that the intersection and union of two algebraic sets is analgebraic set.
6.1 Ideals
Before we can discuss the properties of elementary affine geometry, we need to establish a shortdictionary of ideals. We will begin with the definition of a general ideal. Then we will definemultiple types of ideals. Based upon these definitions, we will explore some of the uses of ideals inelementary affine geometry.
Definition 30. An ideal I of a commutative ring R is a non-empty subset I of R satisfying thefollowing criteria:
1. The set I is closed under subtraction, and
2. If a ∈ I and r ∈ R, then ar ∈ I.
Example 21. Every commutative ring R (except the zero ring) has at least two ideals: {0} andR. The zero ring is only guarenteed to have one ideal since {0} = R.
Definition 31. For a ring R, an ideal I ⊂ R is proper if I 6= R.
Example 22. The set {a ∈ Z : a = 2(x), for some x ∈ Z} = {a ∈ Z : a is even } is a proper idealof Z.
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Definition 32. For a ring R, an ideal I ⊂ R is maximal if I is proper and any ideal J that containsI is either I or all of R.
Example 23. The set {a ∈ Z : a = 3(x), for some x ∈ Z} is a maximal ideal of Z.
Definition 33. For a ring R, an ideal I ⊂ R is prime if I is proper and whenever ab ∈ I fora, b ∈ R, either a ∈ I, b ∈ I, or both are in I.
Example 24. The set {a ∈ Z[x] : a = x(y), for some y ∈ Z[x]} is a prime ideal of Z[x].
Definition 34. An ideal I of a ring R is radical if xn ∈ I or some n ≥ 1 implies that x ∈ I.Given an arbitrary ideal I of a ring R, the radical of I, denoted Rad(I), is the set of all elements
x ∈ R such that some positive power of x is in I.
Example 25. Let f(x, y) = x2 + y2 − 1 ∈ C[x, y]. The ideal I generated by f is radical.
Now that we have established a good basis for understanding ideals, we will explore how theseideals function within affine geometry.
Theorem 11. Let I be the ideal in k[x1, x2, . . . , xn] generated by a set S ⊂ k[x1, x2, . . . , xn]. Then
V (S) = V (I).
Therefore, every algebraic set is defined by an ideal. The following theorem provides a partialconverse to this result.
Theorem 12 (Hilbert Basis Theorem). If the ring R is a Noetherian ring, then R[x] is a Noetherian
ring.
Therefore, every ideal in C[x1, x2, . . . , xn] is finitely generated. Hence, as every algebraic set inA
n(C) is defined by an ideal, every algebraic set is determined by a finite collection of polynomials.
6.2 Hilbert Nullstellensatz
Not only is every algebraic set defined by an ideal, every algebraic set can be uniquely defined by aradical ideal. In an algebraically closed field k, there is a one-to-one correspondence between pointsof A
n(k) and maximal ideals of k[x1, x2, . . . , xn]. This statement is a consequence of what is calledHilbert Weak Nullstellensatz.
Theorem 13 (Weak Nullstellensatz). Let F be an algebraically closed field and I ⊂ F [x1, x2, . . . , xn]be an ideal. Then I is maximal if and only if there are elements a1, a2, . . . , an ∈ F such that
I = 〈x1 − a1, x2 − a2, . . . , xn − an〉.
Theorem 14 (Strong Nullstellensatz). If F is an algebraically closed field and I is an ideal of the
polynomial ring F [x1, x2, . . . , xn], then I(V (I)) = Rad(I).
Notice 〈x2 + 1〉 in R[x] is maximal. However, it does not correspond to any geometric point inA
1(R) = R, which is not an algebraically closed field. Therefore, we must be careful to specify thatthe field k be an algebraically closed field.
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6.3 The Coordinate Ring
In our study of algebraic geometry, our focus has been on sets of points. Specifically, we have beeninterested primarily in the zero sets of polynomials. The coordinate ring focuses on the functionson these sets of points.
Definition 35. Let V be an algebraic set in An(k). The coordinate ring, denoted O(V ), is the set
O(V ) = {f : V → k | f is a polynomial function}
equipped with pointwise addition and multiplication of functions.
Theorem 15. Let V be an algebraic set in An. The following statements are equivalent.
1. The algebraic set V is an affine variety.
2. I(V ) is a prime ideal in k[x1, x2, . . . , xn].
3. The coordinate ring of V , O(V ), is an integral domain.
6.4 Zariski Topology
Before we can discuss the Zariski topology, we must recall the definition of topology.
Definition 36. A topology on a set X is a collection T of subsets of X, which contains the following:
1. The set X,
2. The empty set ∅,
3. The union of any elements of T , and
4. The finite intersection of any elements of T .
In Section 3, we switched from working with the real plane to the complex plane to the projectiveplane in order to prove that all smooth conics are equivalent under change of coordinates. Eachnew plane required a modification of our definition of points. In each of these cases, we defined thecorresponding planes in terms of the points and then algebraic sets in terms of suitable polynomialsexternal to the plane and its points. From this we obtain the coordinate ring of the algebraic set. InZariski’s treatment of algebraic geometry, it is seen that the points, algebraic sets, and polynomialsfunctions can all be reconstructed from the coordinate ring of the algebraic set. Therefore, we canassociate to any commutative ring its corresponding algebraic geometric structure.
Let R be a commutative ring. We need to specify the set of “points” that will be associated tothis ring’s algebraic geometry.
Definition 37. The prime spectrum or spectrum of a ring R, denoted Spec(R), is the collection ofprime ideals in R.
Thus, for a ring R, we obtain the associated space of its prime ideals. This set of points is givena geometric structure by defining its Zariski topology.
Definition 38. Let S ⊆ R for ring R. The Zariski closed set given by S in Spec(R) is Z(S) = {P ∈Spec(R) : P ⊇ S}. A subset U is a Zariski open set if there is a set S ⊆ R with U = Spec(R)−Z(S).
Theorem 16. The Zariski open sets in Spec(R) form a topology.
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Definition 39. A prime ideal I ∈ Spec(R) is Zariski closed if Z(I) = {I}.
Notice that definition does not require all points of Spec(R) to be Zariski closed, since we areworking with prime ideals and not maximal ones. In fact, when you close some of the points, theclosure is the whole set.
Theorem 17. An ideal I ∈ Spec(R) is Zariski closed if and only if I is maximal.
Throughout our brief introduction to algebraic geometry we transitioned from classical algebraicgeometry by depending increaingly more on algebra, specifically ring theory. This transition, justlike our transition from the real plane to the projective plane, caused us to gain power but losegeometric visualization. Thus, with our final step from the affine plane to Spec(R) we sacrifice ourgeometric intuition of the nature of points and visualization for the benefit of the power and toolsof algebra.
7 An Introduction to Algebraic Geometry: A Problem Solving
Approach
The book Algebraic Geometry: A Problem Solving Approach [3] is a preprint text in the processof being written by the Park City Mathematics Institute 2008 Undergraduate Faculty Program.It is written as a book of problems, such that the reader will gain knowledge and understandingthrough the completion of exercises.
The book aims to be applicable to three audiences. The first audience consists of students whohave taken courses in multivariable calculus and linear algebra but have not taken an abstractalgebra course. The second audience consists of students who have taken a course in abstractalgebra. The third audience is referred to as “mathematicians on an airplane.” This is an audienceof mathematicians who would like to learn more about algebraic geometry.
As an undergraduate mathematics major, I have taken a variety of mathematics courses includ-ing calculus, differential equations, real analysis, linear and abstract algebra, topology, probability,mathematical modeling, number theory, non-Euclidean geometry, and Lie groups. With this back-ground, I am a member of the second audience. My experience with algebra and topology wereparticularly helpful for the understanding of the text. Due to my mathematical background, mostof the terms within the book were familiar.
7.1 Experience Editing and Revising the Book
The authors have worked hard, but whenever there is a collaboration or one begins a large project,such as this text, there will be problems along the way. I have separated these current errors andissues into two main categories: issues due to collaboration and issues due to the magnitude of theproject.
The issues due to collaboration consist of the varying writing and formatting styles of eachauthor and the repetition of material and questions. Different styles plague the text. Often thedivision in the text between authors is quite evident. Ideally one would want a smooth continuousflow throughout the book. This goal is not possible without continuity of style. Some examplesof the issues that became evident throughout the text were punctuation inconsistencies, standardwriting style, and standard notation. Punctuation inconsistencies could manifest itself in the place-ment of a comma or the use of a semicolon. The issue may not seem to be large, but I found theinconsistency detracting from the impact of the book. The lack of a standard writing style mani-fested itself in the different writing or teaching styles of each author. Some authors prefer to ask
22
hypothetical questions throughout the text, while others prefer straight sentences. Some authorsprefer to introduce each topic thoroughly giving the reader a little understanding of the topic toserve as a foundation before teaching the details of the topic, while others prefer to dive right oninto the topic at hand. Standard notation for an object within the text varies from author toauthor as well. The other issue due to collaboration is the repetition of material and questions.Since the authors are working independently of each other, they do not know what problems ortopics have been covered by the other authors. While each author has a specific topic to coverfor each section of the text, the author can expand on that topic to include information which hedeems relevant. The issue lies in the fact that different authors may extend their topic to the samerelevant information. This creates the repetition of material within the text.
The issues due to magnitude of project consist of the range of the intended audience and gram-matical and typographical errors. The book aims for an audience ranging from the undergraduatestudent with linear algebra and multivariable calculus to mathematicians wanting to learn about al-gebraic geometry. This leads to the problem of classifying which definitions the audience is assumedto know. For example, the author expects the reader to know abstract algebra at the beginningof Chapter 4. However, the first intended audience does not necessarily have this knowledge. Theother issue due to the magnitude of the project is grammatical and typographical errors. All ofthe authors understand basic grammar rules. However, when one embarks on a large project,grammatical and typographical mistakes are often made.
While we addressed and identified these issues within the book, we also sought to reorder andorganize the content and problems. Sometimes the progression of thought ceased to be linear,rather it jumped about between topics and goals. Our aim was to reorganize some of the materialto provide the reader with a text which flowed logically from one topic to the next.
7.2 Comments on the Book
Based on my experience with the book, I will present a few comments.First, I will address the appropriate audience for the book. The use of such a large range for the
audience causes problems for not only the author but also the reader. An undergraduate withoutabstract algebra cannot be expected to know and understand ring theory. However, if the book waswritten for that undergraduate, then the mathematician reading the book would want to skip largeportions of the book. In order to accommodate this range, the book may have to be broken up intotwo sections: one section would be Chapter 1 through Chapter 3 and the other section would startat Chapter 4. Unlike Chapter 4, the first three chapters of the book expect little background fromthe reader. If the authors choose not to divide the book into two sections, I believe the audienceshould be narrowed. This narrowing will lead to increased availability for the intended audience,and it will aid the authors by supplying a more accurate expectation of the audience’s knowledge.With this increased accuracy, the authors could define terms accordingly and increase the ease oftransition between Chapters 3 and 4 of the text.
Second, I will address the effectiveness of the chosen teaching style, which is a problem-orientedapproach. Personally, I enjoy the approach as a whole and find it to be a quite effective teachingstyle in general. I do think that this approach is appropriate for this text as well. Occasionally,the stating of a definition or a theorem alone can produce confusion. However, the definition ortheorem paired with several examples quickly eliminates this confusion.
The step by step approach to processes or theorems used within the problem-oriented approachallows the reader to concentrate on each step separately to promote long-lasting comprehensionof the concept. In the process of understanding the Weierstrass normal form, we provided aspecific example of a smooth homogeneous cubic curve being transformed into another projectively
23
equivalent smooth cubic curve f1(x1, y1, z1). Then we continue with this example of f1(x1, y1, z1)written in the form αx3
1 + z1F (x1, y1, z1). We show that this specific example can be transformedinto a new polynomial f2(x2, y2, z2) = 0 with the coefficient of x3
2 being 1. Working in the affinepatch z2 = 1, we then transformed a specific example of a non-homogeneous polynomial intoWeierstrass normal form using an affine change of coordinates. We generalized this finding. Afterestablishing the procedure of writing cubics in Weierstrass form, we used that procedure on specificcubics and characterized smooth cubics via invariants related to the Weierstrass normal form. Byestablishing the procedure in this way, the reader is forced to think critically about each step inthe construction of the Weierstrass normal form.
My overall impression of the book is that it is clearly a work in progress, which must undergo alot of editing. Executive decisions have to be made and followed by all of the authors. The ideas andmathematical building blocks are there. The authors only need the addition of proper grammarand style, continuous transitions, and a stable definition of audience to construct an applicablealgebraic geometry text.
8 Conclusion
I gave an brief introduction to elementary algebraic geometry. Section 3 to Section 5 focusedon classical algebraic geometry. Specifically we focused on curves, which included conics, cubics,and higher degree curves. We also explored elementary affine geometry, in which we used ringtheory to describe the geometric features of curves while not being constrained by the real orcomplex numbers. After this brief introduction to the basics of algebraic geometry, I discussed myexperience editing and revising the book along with some of the current issues within the book.Then we moved on to some evaluative comments, including my overall impression of the book.
As the standard undergraduate mathematics major curriculum moves away from the traditionalcurriculum, courses that were once reserved for graduate students are more frequently being offeredto undergraduate students. Algebraic geometry is one of these courses that was once considered ex-clusively a graduate course. As algebraic geometry is offered more frequently at the undergraduatelevel, the need for texts aimed at undergrauate students has increased. David Cox, John Little, andDonal O’Shea met this need with the book Ideals, Varieties, and Algorithms: An Introduction to
Computational Algebraic Geometry and Commutative Algebra [2]. This book uses a computationalapproach to teaching the subject to undergraduate audiences.
The book written by the Park City Mathematics Institute 2008 Undergraduate Faculty Pro-gram [3] aims to teach the classical material of algebraic geometry using a contemporary, problem-oriented method. This approach encourages critical thinking and interactive learning. Therefore,Algebraic Geometry: A Problem Solving Approach [3] aims to provide a text written in an accessiblemethod that will continue to meet the recent need for undergraduate algebraic geometry texts.
References
[1] Anderson, Marlow; Feil, Todd. A First Course in Abstract Algebra. Second edition. Chapman& Hall/CRC, Boca Raton, FL, 2005.
[2] Cox, David; Little, John; O’Shea, Donal. Ideals, Varieties, and Algorithms. An Introduction to
Computational Algebraic Geometry and Commutative Algebra. Third edition. UndergraduateTexts in Mathematics. Springer, New York, NY, 2007.
[3] Garrity, Thomas et al. Algebraic Geometry: A Problem Solving Approach. Preprint. 2009.
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[4] Lang, Serge. Introduction to Algebraic Geometry. Third printing, with corrections. Addison-Wesley Publishing Co., Reading, MA, 1972.
[5] Reid, Miles. Undergraduate Algebraic Geometry. London Mathematical Society Student Texts,12. Cambridge University Press, Cambridge, 1988.
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