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

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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

Flattening Stratifications

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

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

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

Posted in AG From the Beginning, Algebraic Geometry, Cohomology, Hilbert Scheme | 1 Comment

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