Monthly Archives: January 2008

Papers, Readers and More

Posted on January 31, 2008 by Charles Siegel

Today, rather than posting in my Algebraic Geometry from the Beginning series, I’m just going to do a quick post on a few things that are going on mathematically. First off, by clicking through Kung Fu Physics’s Super Holiday Post … Continue reading →

Posted in Algebraic Geometry | 1 Comment

Morphisms of Sheaves

Posted on January 30, 2008 by Charles Siegel

We’ve got sheaves now, so naturally we move on to morphisms between them. We begin by fixing a topological space and take to be two presheaves on . A morphism of presheaves is an abelian group homomorphism for each open, … Continue reading →

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

Sheaves

Posted on January 29, 2008 by Charles Siegel

Second in our survey of some technical tools are sheaves. Sheaves have a reputation for being terrifying and technical and some people I know have trouble seeing their purpose at all. In my opinion, they’re very nice, convenient ways to … Continue reading →

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

Localization

Posted on January 29, 2008 by Charles Siegel

Now that I’m settling into the semester, it’s time that I get back to rambling semi-coherently about algebraic geometry. I do, however, now have a somewhat more focused goal, and though it’s still quite a ways off, it has reminded … Continue reading →

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

Tangent Spaces and Singular Points

Posted on January 23, 2008 by Charles Siegel

The semester is starting up again now, so my schedule is a bit irregular. Sadly, I won’t be able to keep up the pace I set over break. However, I should still be putting out a couple of posts a … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry | 8 Comments

Grassmannians and Flag Varieties

Posted on January 17, 2008 by Charles Siegel

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

Posted on January 16, 2008 by Charles Siegel

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

The Veronese Embedding

Posted on January 14, 2008 by Charles Siegel

Now we’ve got some basic theory under our belts. Time to get a handle on specific objects. Nothing gives better intuition into algebraic geometry than working out a bunch of examples and seeing how they fit together. We’ll begin with … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Examples | 12 Comments

Elliptic Curves

Posted on January 11, 2008 by Charles Siegel

We’ve now spoken about all sorts of invariants, the most important ones being derived from the Hilbert Polynomial, we’ve spoken about Bezout’s Theorem, and we’ve developed a nice bit of classical algebraic geometry. We’ve also mentioned algebraic groups, and it … Continue reading →

Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Curves, Examples | 3 Comments

Bezout’s Theorem

Posted on January 10, 2008 by Charles Siegel

Now we’re ready for our first big geometric theorem. We’ll prove Bezout’s Theorem as a special case of another, more general theorem.

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Curves, Intersection Theory | 9 Comments