Read What is Algebraic Geometry?

Adapted by Dr. Sarah from excerpts taken from Andreas Gathmann's What is algebraic geometry?

What is algebraic geometry? To start from something that you probably know, we can say that algebraic geometry is the combination of linear algebra and algebra:

Algebraic geometry combines these two fields of mathematics by studying systems of polynomial equations in several variables. Given such a system of polynomial equations, what sort of questions can we ask? Note that we cannot expect in general to write down explicitly all the solutions. So we are more interested in statements about the geometric structure of the set of solutions. For example, in the case of a complex polynomial equation of degree d, even if we cannot compute the solutions we know that there are exactly d of them (if we count them with the correct multiplicities). Let us now see what sort of "geometric structure" we can find in polynomial equations in several variables.

Let's look at the polynomial equation in two variables

y

We can look at the set of solutions (x,y) that satisfy this equation. Recall that the complex plane is made up of numbers a + b i, where i is the square root of -1.

Let's go back to y

So it seems that the set of solutions looks like two copies (a,b), (a,-b) of the complex plane with the two copies of each point 1,...,2n identified. While this is not quite true, the actual behavior is beyond the scope of this course (since the complex analysis is more difficult). So, the actual behavior is a more difficult, but we can use our naive understanding to visualize the set of solutions.

We start with two copies of the complex plane, and cut out holes to (roughly) represent the roots. We then glue the two copies of the plane together along these holes. This gives us a geometric picture of the set of solutions to this equation.

Answer questions 1 and 2 for this equation.

One of the trivial but common sources for misunderstandings is whether we count dimensions over the complex numbers (C) or the reals (R). We obtain complex curves (the dimension over C is 1) or real surfaces (the dimension over R is 2). This should help clarify the terminology from "The Proof" video that an elliptic curve (complex 1D curve that arises from a cubic, or degree 3, equation y

Let's look at a more complicated object, namely complex surfaces in 3-space (x, y, z can be complex). Look at the equation

1 + x

In contrast to our previous examples, we have now used a linear projection to map the real 3-dimensional space onto the drawing plane. We see that there are some lines contained in S. In fact, one can show that every smooth cubic surface has exactly 27 lines on it. This is another sort of question that one can ask about the solutions of polynomial equations, and that is not of topological nature: do they contain curves with special properties (in this case lines), and if so, how many? This branch of algebraic geometry is usually called enumerative geometry. |

What we should learn from the examples above:

Let's look at the example y

Read What can algebraic geometry be used for?

If time remains, then skim through The Algebraic Geometry Notebooks for Non-Experts by Aksel Sogstad . Note: If you don't have time to read this in class, then don't worry about it, as this is not homework.

Adapted by Dr. Sarah from excerpts taken from Peter Stiller's What is Algebraic Geometry, The Algebraic Geometry Notebooks for Non-Experts, Tyler J. Jarvis' Projections of Complex Algebraic Curves to Real 3-space , and Andreas Gathmann's What is Algebraic Geometry?