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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.

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# Category Archives: Hilbert Scheme

## The Clemens Conjecture

This is a draft of a talk I’m giving on Thursday, mostly going over the work of Katz in this paper. Let’s begin with an informal statement of the conjecture: Conjecture: For every positive integer , a general quintic hypersurface … Continue reading

## 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

## Flattening Stratifications

Now we move on to the next ingredient in the construction: flattening stratifications. We’ll start with just stating the theorem that Kollár used: Theorem: Let be a projective scheme and ample. Let be a coherent sheaf on . For every … Continue reading

## Constructing the Hilbert Scheme II

Last time we did a quick run through of how to put together the Hilbert Scheme. A few questions came up in the comments, the first being: how can we guarantee the existence of the we used, which works uniformly … Continue reading

## 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