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\def\P{\bf P} Given a variety $X$ in projective $r$-space $\P^r$, the Fano scheme $Fano_k(X)$ is the natural parameter space for the linear $k$-planes lying on $X$. In this tutorial we explore the methods for computing it. The tutorial is in three parts

A. The twenty-seven lines

B. General methods

C. Surfaces of degree $4$ in $\P^5$

In section A, we treat by hand the Fano variety of lines on a nonsingular cubic surface in $\P^3$, and find that there are indeed $27$ lines lying on the surface.

In section B, we explain a general purpose function, written to compute Fano schemes.

There are (up to linear transformations) just $3$ nondegenerate smooth surfaces of degree $4$ in $\P^5$: the Veronese embedding of $\P^2$ and the rational normal scrolls $S(1,3)$ and $S(2,2)$. Can they be distinguished by their Fano varieties of lines? To find out, read section C ! \beginsection{ A. Lines on the nonsingular cubic in $\P^3$}\par First make the homogeneous coordinate ring of the ambient projective $3$-space

and the ideal of a nonsingular cubic

We make a parametrized indeterminate line in our projective space, adding parameters $s,t$ for the line and two points $p_0..p_3$ and $q_0..q_3$ representing the points 0 and infinity on the line.

Then we make a map $F$ from $R$ to the new ring sending the variables to the coordinates of the general point on the line

We now apply $F$ to the ideal of $X$

and the condition we want becomes the condition that {\tt FX} vanishes identically in $s,t$. The following line produces the coefficients:

The interface to the {\tt coefficients} routine is a bit baroque, and might change in the future. For now, the $\{0,1\}$ says to find the coefficients of each column of the matrix, with respect to the first two variables. The routine returns a list of two matrices, the second one being the one we need (index 1, since all indices start at 0 in Macaulay2)

We can get rid of some of the variables of $S$, to ease the computation:

The ring we want is the quotient

Now we want to move to the Grassmannian, so we take a new polynomial ring in 6 variables that will correspond to the minors of the matrix with rows $p_0..p_3$ and $q_0..q_3$,

We define a map sending the $x_i$ to the minors, regarded as elements of S1bar

{\tt trim} replaces the given set of generators with a minimal set of generators. We get an ideal representing points:

and the number of these points -- the number of the corresponding lines - is 27:

It is interesting to note that the ideal of the Fano scheme that we have produced is NOT saturated, as the number (25) of cubics it contains is less than $56 - 27 = 29$:

Possible next steps in this computation would be to compute the Jacobian ideal of Fano to show that we really got 27 distinct lines, etc. \beginsection{ B. General methods}\par The first step in writing a program is to decide what the program should do, and it is just as well to write the documentation at this point .

First, Fano and Grassmannian are commands in Macaulay 2, so are a protected symbol. We remove its current meaning by

The documentation has the following form:

If we take the variety that is the whole of $\P^r$, we get the Grassmannian. It is useful to be able to make the ring representing the ambient space of the Grassmannian beforehand by hand, so the ideals of several Fano varieties can be compared. But often we won't need this. Thus we make the function capable of accepting this ambient ring as an argument, or of fending for itself if no ambient ring is given.

In order to make {\tt Fano} handle an optional number of arguments, we make it a method instead of a function, as follows

Here is the code for the first case, with comments interspersed:

The second case reduces to the first:

With the 0 ideal we get the Grassmannian of projective $k$-planes in $\P^r$:

As a first example we can try the Fano of lines on the nonsingular quadric in $\P^3$

we investigate by checking its dimension and degree

The answer ``2'' means that $I$ is the ideal of a curve in $\P^5$, the ambient space of the Grassmannian of lines.

The answer is 4. In fact, the ideal $I$ represents the union of two conics. \beginsection{ C. Surfaces of degree $4$ in $\P^5$}\par We now turn to the three surfaces of degree $4$ in $\P^5$, and make their ideals:

The ring of $\P^5$

It happens that the ideals of all three surfaces are generated by minors of suitable matrices:

The Veronese embedding of $\P^2$:

The other scrolls are defined by the minors of matrices that are made from ``catalecticant'' blocks, that is, from matrices such as $$\begin{pmatrix}b & c & d & e \\ c & d & e & f \end{pmatrix}$$ which are manufactured by

for example

produces the example above. The rational normal scroll {\tt S13}, which is the union of lines joining a line with the corresponding points of a twisted cubic in a disjoint subspace of $\P^5$

Finally, the rational normal scroll {\tt S22}, which is made by a similar construction starting with two conics in $\P^5$

It is interesting to note that the numerical invariants of these surfaces are very hard to distinguish. In particular, the graded betti numbers

coincide, so the three cannot be distinguished on the basis of these or on the basis of the (weaker) invariants the Hilbert series or Hilbert polynomials. But the Fano varieties are more obviously different:

We compute the Fano varieties of lines on each of our surfaces.

The ideal contains all $120$ quadrics, and represents the empty set: The Veronese surface contains no lines!

It turns out that the dimension (1) and degree (4) of these varieties coincide! Moreover, since the ideals are not saturated, one cannot directly compare the Hilbert series or free resolutions (of course one could first compute a saturation). But there is the arithmetic genus, that is, $1-H(0)$, where $H$ is the Hilbert polynomial.

The output, $4 \P^1 - 2 \P^0$, means ``four times the Hilbert polynomial of the projective line minus 2''; that is, the polynomial is $H(d) = 4d + 2$; so arithmetic genus is $-1$.

The output, $4 \P^1 - 3 \P^0$, means $H(d) = 4d + 1$, arithmetic genus $-1$. In fact, the Fano variety of {\tt S22} consists of a projective line, embedded as a smooth rational quartic; while the Fano variety of {\tt S13} consists of a smooth rational quartic (corresponding to the rulings of the surface {\tt S13} and an isolated point, corresponding to the section of negative self-intersection on the surface).

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