Subsection 5.4.3
To treat tangent bundles to hypersurfaces in Schubert2, we have to be a little more careful. If $X$ is a hypersurface in ${\mathbb P}^n$, we cannot hope to construct the Chow ring to $X$. Even for the case of an elliptic curve $E$ (a degree-3 hypersurface in $\mathbb{P}^2$), the construction of $A^1(E)$ amounts to completely understanding the group law on $E$ and all points of $E$ (so in particular, this ring is never finitely generated over $\mathbb{C}$), and the situation quickly gets worse for higher dimensions and degrees.
However, for classes on $X$ which are obtained by restricting classes on ${\mathbb P}^n$ to $X$, we can easily understand a great deal via the projection formula, which in this particular case tells us that if $i:X \rightarrow {\mathbb P}^n$ is the inclusion, then
$$i_*(\alpha|_X) = \alpha \cap [X]$$
So, if for example we are interested in calculating the degree of $\alpha|_X$, we can equivalently calculate the degree of $\alpha \cap [X]$. In this way we ``push the problem forward'' to ${\mathbb P}^n$.
As an example, if we want to calculate the degree of the top Chern class of the tangent bundle to a hypersurface $X$ of degree $4$ in ${\mathbb P}^3$, we can compute:
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This works because we have $$c(T_X) = \frac{c(T_P)|_X}{c(N_X)} = \frac{c(T_P)}{c(O_P(X))}|_X.$$
More generally, we can compute the Euler characteristic of a degree-$d$ hypersurface in $\mathbb{P}^n$ as in the book. We can even compute a closed formula for all $d$ and fixed $n$ using base.
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And now we can similarly calculate a formula for the middle Betti number of such a hypersurface:
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Using this, we easily reproduce the table given in the text:
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Exercise 5.11: Betti numbers of smooth complete intersections
In the same way as for hypersurfaces, we compute that if $X$ is a complete intersection of hypersurfaces of degrees $d_1, \ldots, d_k$ in $P = {\mathbb P}^n$, then $$c(T_X) = c(T_P)/(\prod_{i=1}^k c(O_P(d_i)))|_X$$ We can use then Schubert2 to produce a closed-form formula for the degree of the top Chern class of the tangent bundle to a complete intersection of $k$ hypersurfaces in ${\mathbb P}^n$:
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And from here we can compute the middle Betti numbers:
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Now our particular problem is easy:
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For good measure, we'll also compute the Euler characteristics:
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Exercise 5.12: Betti numbers of the quadric line complex
The only interesting Betti number is the middle one, which we compute immediately from the above:
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Exercise 5.13: Calculate the Euler characteristic of a smooth hypersurface of bidegree $(a,b)$ in ${\mathbb P}^m \times {\mathbb P}^n$
Working on ${\mathbb P}^m \times {\mathbb P}^n$ in Schubert2 is easy using relative flag bundles (or relative projective spaces): this space is the same as the projectivization of a trivial rank-$n+1$ bundle on ${\mathbb P}^m$. So, for example, to build ${\mathbb P}^2 \times {\mathbb P}^3$:
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Note that if we didn't use the VariableNames options this ring would be horrible to look at, since classes pulled back from ${\mathbb P}^2$ and ${\mathbb P}^3$ would both be named $H$.
We can calculate a closed-form formula for the Euler characteristic of a smooth hypersurface of bidegree $(a,b)$ once we have fixed $m$ and $n$, but we cannot use $m$ and $n$ as base parameters themselves.
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The source of this document is in Book3264Examples.m2:783:0.