Posting is slowing down a bit, I’ve got a paper I’m trying to get out, and a couple of projects that are hitting some preliminary results, plus, I’m getting ready for holiday travel, and then two months at Humboldt.  Trying out an experiment with more rigid personal scheduling, and hopefully I’ll post more often.  Also, I’m reviewing Atiyah-Macdonald, Eisenbud, and Schenck so that perhaps in March I can begin a “Commutative Algebra from the Beginning” series, or perhaps just a series on geometric interpretation of commutative algebra theorems.

However, for today, we’re going to take something most of us first saw in differential geometry (I first met this map in do Carmo‘s book) and translate it into algebraic geometry.

We will start in the absolute least general way possible, following do Carmo.  Let S\subset \mathbb{R}^3 be a surface.  Then there’s a map N:S\to \mathbb{S}^2 to the unit sphere taking each point to its unit normal vector.  This is the Gauss map, and it’s a REALLY useful tool, as anyone who has gone through this book can attest.  For instance, if you want to define the curvature of a surface in \mathbb{R}^3, the Gauss map is essential.  For instance, the Gaussian curvature is the determinant of the derivative of N, and in fact it would be redundant to go through everything about the Gauss map for surfaces in \mathbb{R}^3 because there’s a whole chapter in do Carmo titled “The Geometry of the Gauss Map!”

We’re going to generalize and then algebraize.  First, let’s just drop the orientation on our surface.  To forget that, we can replace the normal vector with the normal line.  So then instead of getting a point in \mathbb{S}^2 we get a pair of antipodal points, or just a point of \mathbb{RP}^2 from our surface.  Then we can see that the map is really given by taking \iota:S\to \mathbb{R}^3 the inclusion, then we have d\iota_p:T_p(S)\to T_p(\mathbb{R}^3), and then taking the line perpendicular to the image plane.  Taking the union of these maps, we just have the map d\iota:TS\to T\mathbb{R}^3.  Then, using the Riemannian metric on \mathbb{R}^3, we can make this a map N:N_{S/\mathbb{R}^3}\to T\mathbb{R}^3, and follow it up with the fact that T\mathbb{R}^3\cong \mathbb{R}^3\times\mathbb{R}^3, and project down, to get the map N_{S/\mathbb{R}^3}\to \mathbb{R}^3, and then we can rewrite it by taking each point to the line in \mathbb{R}^3, which gives us a map S\to \mathbb{RP}^2, giving the usual Gauss map.

So how can we simplify and generalize this? Step 1 is to replace the normal vector with the tangent space, which gives a point in the dual projective space.  Then we want to generalize dimension.  It’s easy to handle hypersurfaces in \mathbb{R}^n, we just get a map to \mathbb{RP}^{n-1} (or rather, to its dual).  In general, if we allow non-hypersurfaces, we get maps to Grassmannians, so M^k\to \mathbb{R}^n gives us a Gauss map N:M\to Gr(k,n).

Now, we’re going to let the target space vary.  We just need a space Y such that TY\cong Y\times V where V is the tangent space at some specific point.  If we have trivial tangent bundle, we can identify all the fibers and then the derivative of our map actually gives us a map from the domain to a Grassmannian.  What are some spaces that have this property? Lie groups! It’s important that we have Lie groups, not just homogeneous spaces, because of the unique way that we can identify fibers.

Now, if we try to algebraize, the first thing we get is a Gauss map for affine varieties X\to \mathbb{A}^n.  We can even get rid of \mathbb{A}^n and replace it with an algebraic group G, but most of the algebraic groups that immediately come to mind are affine, things like GL(n), SO(n), Sp(n) etc, the classical groups.  Plus \mathbb{G}_m and \mathbb{G}_a and products of these.  But that still only gives us affine varieties, nothing projective or complete.  Fortunately, there’s one remaining option that are commonly studied: abelian varieties.  Though we can’t do much with rational varieties (as there are no maps from them into abelian varieties other than constant maps), we can get a lot of mileage out of the Gauss map on abelian varieties, as we’ll see in the next post.

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