# carpet -- Ideal of the unique Gorenstein double structure on a 2-dimensional scroll

## Synopsis

• Usage:
I = carpet(a1,a2)
I = carpet(a1,a2,m)
(I,xmat,ymat) = carpet(a1,a2,Scrolls=>true)
• Inputs:
• a1, an integer,
• a2, an integer, a1 and a2 should be positive
• m, , a 2xn matrix for some $n \ge{} a1+a2$
• Optional inputs:
• Characteristic => an integer, default value 32003, the characteristic of the ground field
• Scrolls => , default value false, if true return in addition the matrices defining the sections
• FineGrading => , default value false, if true then I is defined over the ring with $\ZZ^4$-grading
• Outputs:
• I, an ideal,
• xmat, ,
• ymat, , the matrices of the sections of the scroll
• Consequences:
• If no matrix m is present then the script creates a type a1,a2 K3-carpet over a new ring. If m is given, then an ideal made from certain minors and sums of minors of m is produced. The characteristic is given by the option, defaulting to 32003. If the option FineGrading is set to true, then the ideal is returned with the natural $\ZZ^4$ grading (the default is FineGrading => false). This last may not work unless the matrix is of scroll type (or not given!) If Scrolls=>true, then a sequence of three items is returned, the second and third being the smaller and larger scroll matrices.

## Description

The routine carpet(a1,a2,m) sets a = min(a1,a2), b = max(a1,a2), and forms two matrices from m: X:the 2 x a matrix that is the first a cols of m; Y:the 2 x b matrix that is the nex b cols of m–that is, cols a1..a1+a2-1 of m; Let Ix, Iy be the ideals of 2 x 2 minors of X and Y. If $a,b\geq 2$,the routine returns Ix+Iy+Imixed, where Imixed consists of the quadrics "outside minor - inside minor", that is, $det(X_{\{i\}},Y_{\{j+1\}})-det(X_{\{i+1\}}|Y_{\{j\}})$, for each pair of (i,i+1), (j,j+1) in the ranges a1 and a2.

If m is usual ideal of the scroll of type (a,b), then carpet(a,b,m) produces the same ideal (over a different ring) as carpet(a,b). This is the ideal of the 2-dimensional rational normal scroll Scroll(a1,a2) is the ideal of 2 x 2 minors of X|Y. The ideal I to be constructed is the ideal of the unique (numerically) K3 scheme that is a double structure on the scroll S(a1,a2).

When a,b > 1, the carpet ideal I is the sum $Ix+Iy$ plus the ideal Imixed

When a = b = 1, I is the square of the determinant of X|Y.

When a = 1, b>1 (or symmetrically), I is defined as in the case a,b>1, after replacing $$X = \begin{pmatrix} x_0 \\ x_1 \end{pmatrix}$$

by the 2 x 2 matrix $$\begin{pmatrix} x_0^2 & x_0*x_1 \\ x_0*x_1 & x_1^2 \end{pmatrix}$$ and changing $a$ to 2.

 i1 : betti res carpet(2,5) 0 1 2 3 4 5 6 o1 = total: 1 15 49 70 49 15 1 0: 1 . . . . . . 1: . 15 35 35 14 . . 2: . . 14 35 35 15 . 3: . . . . . . 1 o1 : BettiTally i2 : S = ZZ/101[a..j] o2 = S o2 : PolynomialRing i3 : m = genericMatrix(S,a,2,5) o3 = | a c e g i | | b d f h j | 2 5 o3 : Matrix S <--- S i4 : I = carpet(2,3,m) o4 = ideal (b*c - a*d, b*e - a*f, d*e - c*f, d*g - c*h - b*i + a*j, f*g - e*h ------------------------------------------------------------------------ - d*i + c*j, h*i - g*j) o4 : Ideal of S i5 : L = primaryDecomposition I; i6 : betti res L_0 0 1 2 3 4 o6 = total: 1 10 20 15 4 0: 1 . . . . 1: . 10 20 15 4 o6 : BettiTally i7 : betti res L_1 0 1 2 3 4 5 o7 = total: 1 15 40 45 24 5 0: 1 . . . . . 1: . 15 40 45 24 5 o7 : BettiTally

## Caveat

We require $a1,a2 \ge 1$. If $a1>a2$ then the blocks are reversed, so that the smaller block always comes first. The script generalizeScroll is a more general tool that can do the same things.