Category Archives: Hilbert Scheme

The Clemens Conjecture

Posted on February 1, 2009 by Charles Siegel

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 →

Posted in Algebraic Geometry, Curves, Enumerative Geometry, Hilbert Scheme, Intersection Theory, Talks | Leave a comment

Examples of Moduli Spaces

Posted on July 23, 2008 by Charles Siegel

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

Posted on July 22, 2008 by Charles Siegel

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 →

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

Constructing the Hilbert Scheme II

Posted on July 21, 2008 by Charles Siegel

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

Posted on July 18, 2008 by Charles Siegel

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