Category Archives: Examples

Constructing Nodal Curves

So, I don’t really have a full length post in me at the moment. However, here’s a nice trick that I’ve learned recently, plus some motivation. With this, it’s easy to write down explicitly a curve of arbitrary degree in … Continue reading

Posted in Algebraic Geometry, Curves, Examples | 11 Comments

Group Schemes and Moduli (I)

Hi! My name is Matt DeLand, I’m a graduate student at Columbia and I’m responding to Charles’ call for cobloggers. I also study Algebraic Geometry, and have been enjoying Charles’ posts; hopefully I can help out and make some positive … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Examples | 5 Comments

Canonical Linear Systems

Given any curve , we have a natural linear system , consisting of all the effective canonical divisors. Now, sometimes this is unhelpful. After all, if , then the canonical system has negative degree, so . If , then the … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Curves, Examples | 5 Comments

Examples of Moduli Spaces

Now we’re done constructing , so it’s time to get the general Hilbert scheme done, and then to construct some other moduli spaces. Now, as , we can see that is a subfunctor of , so we want to try … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Deformation Theory, Examples, Hilbert Scheme | 3 Comments

The Hilbert Scheme

Now that we have a notion of moduli space, we’re going to look at several concrete examples. I mentioned that the Grassmannian is a fine moduli space for -planes in . We’ll make use of this to construct several more. … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Deformation Theory, Examples, Hilbert Scheme | 17 Comments

Grassmannians and Flag Varieties

Back in the second post, we defined projective space to be the collection of lines through the origin in affine space. A natural generalization is to look at -planes through the origin in affine space. At first glance, we might … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Examples | 4 Comments

The Segre Embedding

Today, we talk about the Segre Embedding. This will let us say a bit more about the products of varieties that were mentioned before.

Posted in AG From the Beginning, Algebraic Geometry, Examples | 9 Comments