Varieties of triangles

In section 2.5 we study varieties of triangles. As mentioned above is smooth for an arbitrary smooth projective variety X. So we can use the Weil... 

Let X be a smooth projective variety of dimension d over a field k. For d > 3 and n > 4 the Hilbert scheme Xl" is singular. However X 3 is smooth for all d IN. In this section we want to compute the Betti numbers of X can be viewed as a variety of unordered triangles on X. We also consider a number of other varieties of triangles on X, some of which have not yet appeared in the literature. As far as this is not yet known, we show that all these varieties are smooth. We study the relations between these varieties and compute their Betti numbers using the Weil conjectures. 

This means we consider Hilb (X) as a variety of triangles with a side and the opposing vertex marked. Let... 

Hilb (X) the zero-cycle ujn(Zn) — w -i(2 i) is [x] for some point x g X and res(Zn-i, Zn) — x. If we consider Hilb (X) as a variety of triangles with a marked side, then res maps such a triangle to the vertex opposite to the marked side. 

This morphism is birational, as its restriction gives an isomorphism from (Hilb ( ))(i,i,i) to a dense open subset of Zi(X). So pi3 Hilb (X) — Zs(X) is a canonical resolution of Z3(X). We can consider Z3(X) as the variety of triangles with a marked vertex. Then plt3 is given by forgetting the marked side. 

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