Subsection 4.2.2
Schubert2 identifies ${\mathbb G}(1,3)$ with its Chow ring. We can see this directly using the command intersectionRing.
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The generators $H_{i,j}$ of the above ring are defined by the formula $H_{i,j} = c_j(B_i)$ where $B_i$ is the i-th bundle in the list G.Bundles (numbered starting with 1) and $c_j$ is the j-th Chern class, defined in Ch. 5. The relationship with the Schubert classes on ${\mathbb G}(1,3)$ is as follows:
$H_{2,1} = \sigma_1$
$H_{2,2} = \sigma_2$
$H_{1,1} = -\sigma_1$
$H_{1,2} = \sigma_{1,1}$
The Schubert classes can also be accessed directly using the placeholderSchubertCycle command -- see Section 4.3 for details.
As an example, we can compute $(\sigma_1)^2$:
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Oops! This just gave us $H_{2,1}^2$ back! Schubert2 actually uses $\sigma_1^2$ and $\sigma_{1,1}$ as its "preferred basis" for the codimension-2 part of the Chow ring of ${\mathbb G}(1,3)$. To convert to the Schubert basis, we use the function placeholderToSchubertBasis:
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We recover the formula of Theorem 4.13: $\sigma_1^2 = \sigma_2 + \sigma_{1,1}$.
Subsection 4.2.4
How many lines in ${\mathbb P}^3$ meet four general lines?
After phrasing the problem in terms of Schubert calculus, this is easy to calculate both by hand and in Schubert2:
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The command integral here returns the degree of the zero-cycle $(\sigma_1)^4$, which is the number we want (namely, 2).
Lines meeting a curve
We can easily build a function which, given the degree $d$ of a space curve $C$, returns the cycle of lines in ${\mathbb P}^3$ meeting $C$:
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And now we can calculate, for example, how many lines meet four general conics:
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But we really want to answer the question once and for all: how many lines meet four general curves of degree $d$? To do this, we use the base command, which allows us auxiliary parameters:
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And we get back the answer $2d^4$, solving the problem once and for all.
Chords to a Space Curve
For each $d$ and $g$ we build the cycle in ${\mathbb G}(1,3)$ of lines secant to a general curve of degree d and genus g:
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The keynote question was: how many lines are secant to two general twisted cubics? But we can do better, and answer the question: how many lines are secant to two general curves of degree $d$ and genus $d$?
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Now if we want to answer our specific question, we just substitute in the desired values for $d$ and $g$:
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WARNING: because of some ugly M2 design decisions, if you don't make at least one of $d'$ or $g$ a rational number, this substitute will return the wrong answer! Hopefully this design will be changed in the future.
Exercise 4.25 (a):
If $C$ is a smooth, nondegenerate space curve and $L$ and $M$ are general lines in ${\mathbb P}^3$, how many chords to $C$ meet both $L$ and $M$? Using our work above, we immediately compute:
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Tangent Lines to a Surface
Exercise 4.28:
Using our Grassmannian G' with an extra base parameter $d$, we build the cycle of tangent lines to a general surface of degree $d$:
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Now we can compute the number of lines tangent to four general surfaces of degree $d$:
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In particular, we calculate the number of lines tangent to four general quadric surfaces:
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The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Book3264Examples.m2:401:0.