### About this blog

Rigorous trivialities is a web log about mathematics, but especially geometry, broadly construed. Contributors will be Charles Siegel, Jim Stankewicz and occasionally Matt Deland. Charles specializes in algebraic geometry, topology and mathematical physics. Jim specializes in arithmetic algebraic geometry. Matt has transitioned from algebraic geometry to work in industry.

Header is taken from the larger work by fdecomite under the creative commons license.

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# Monthly Archives: January 2008

## Papers, Readers and More

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

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

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

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

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

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

## The Veronese Embedding

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

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

## Bezout’s Theorem

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.