This package accompanies our paper [ES18], Equations and syzygies of K3 carpets and union of scrolls, for experimental exploration. There is a unique surjection from the ideal of a 2-dimensional rational normal scroll (other than the cone over a rational normal curve) onto the canonical module of the scroll and the kernel of the this map is the ideal of a scheme that looks numerically like a K3 surface: a K3 carpet. (Theorem 1.3 of Degenerations of K3 surfaces in projective space, by Francisco Gallego and B.P. Purnaprajna, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2477–2492.)## Constructions

## Analyzing

## Correspondence Scrolls

## Relative resolutions of X_e(a,b) in case of k resonance

## Homotopies

The carpet lies on the intersection of the cones over two rational normal curves Ca and Cb of degrees a>=b. We write the ideal of Ca as the minors of a 2xa matrix X with entries x_i, i= 0..a, and similarly for Cb, with a 2 x b matrix Y with entries y_j. We write Xi for the ith column of X, and similarly for Y. In the general case, where a,b are both >=2, the additional generators of the ideal of the Carpet are then given by the differences det(Xi,Yj)-det(X(i+1),Y(j-1)), or equivalently, by the minors of (Xi+Yj,X(i+1)+Y(j-1), (In the case a=1=b the ideal is the square of the determinant of X|Y; if a>1, b=1 then for the mixed minors we replace the 1-column matrix Y by a symmetric 2x2 matrix with entries y_0^2,y_0y_1,y_1^2 )

The hyperplane section of a K3 carpet is a canonical ribbon indexed by genus g=a+b+1 and clifford index b.

The K3 carpets generalize to a family of degenerate K3 surfaces which are unions of two scrolls, whose hyperplane sections are reducible canonical curves consisting of two rational normal curves of degree g-1 intersecting in g+1 points. The functions in this package explore the syzygies of these surfaces for fields of arbitrary characteristic. Inparticular, the functions in the package allow for g <= 15 a computational proof of the following conjecture.

Conjecture 0.1 A general canonical curve of genus g over a field of characteristic p satisfies Green's conjecture, if p >= (g-1)/2.

- carpet -- Ideal of the unique Gorenstein double structure on a 2-dimensional scroll
- canonicalCarpet -- Carpet of given genus and Clifford index
- gorensteinDouble -- attempts to produce a Gorenstein double structure J subset I

- carpetBettiTables -- compute the Betti tables of a carpet of given genus and Clifford index over all prime fields
- carpetBettiTable -- compute the Betti tables of a carpet of given genus and Clifford index over a prime field of characteristic p
- analyzeStrand -- analyze the (a+1)-st constant strand of F over ZZ
- degenerateK3BettiTables -- compute the Betti tables of a degenerate K3 over all prime fields
- schreyerName -- get the names of generators in the (nonminimal) Schreyer resolution according to Schreyer's convention
- allGradings -- add Grading to a chainComplex
- carpetDet -- compute the determinant of the crucial constant strand of a carpet X(a,b)
- resonanceDet -- compute the resonance determinant of the crucial constant strand of a degenerate K3 X_e(a,a)

- correspondenceScroll -- Union of planes joining points of rational normal curves according to a given correspondence
- hankelMatrix -- matrix with constant anti-diagonal entries
- productOfProjectiveSpaces -- Constructs the multi-graded ring of a product of copies of P^1 (pp is a synonym)
- schemeInProduct -- multi-graded Ideal of the image of a map to a product of projective spaces
- smallDiagonal -- Ideal of the small diagonal in (P^1)^n
- irrelevantIdeal -- returns the irrelevant ideal of a multi-graded ring
- degenerateK3 -- Ideal of a degenerate K3 surface X_e(a,b)

- resonanceScroll -- compute the splitting type of the resonance scroll
- coxMatrices -- compute the Cox matrices
- relativeEquations -- compute the relative quadrics
- relativeResolution -- compute the relative resolution
- relativeResolutionTwists -- compute the twists in the relative resolution
- computeBound -- compute the bound for the good types in case of k resonance

- homotopyRanks -- compute the ranks of the quadratic homotopies on a carpet
- canonicalHomotopies -- Homotopies on the resolution of a K3 carpet

- Functions and commands
- allGradings -- add Grading to a chainComplex
- analyzeStrand -- analyze the (a+1)-st constant strand of F over ZZ
- canonicalCarpet -- Carpet of given genus and Clifford index
- canonicalHomotopies -- Homotopies on the resolution of a K3 carpet
- carpet -- Ideal of the unique Gorenstein double structure on a 2-dimensional scroll
- carpetBettiTable -- compute the Betti tables of a carpet of given genus and Clifford index over a prime field of characteristic p
- carpetBettiTables -- compute the Betti tables of a carpet of given genus and Clifford index over all prime fields
- carpetDet -- compute the determinant of the crucial constant strand of a carpet X(a,b)
- computeBound -- compute the bound for the good types in case of k resonance
- correspondenceScroll -- Union of planes joining points of rational normal curves according to a given correspondence
- coxMatrices -- compute the Cox matrices
- degenerateK3 -- Ideal of a degenerate K3 surface X_e(a,b)
- degenerateK3BettiTables -- compute the Betti tables of a degenerate K3 over all prime fields
- gorensteinDouble -- attempts to produce a Gorenstein double structure J subset I
- hankelMatrix -- matrix with constant anti-diagonal entries
- homotopyRanks -- compute the ranks of the quadratic homotopies on a carpet
- irrelevantIdeal -- returns the irrelevant ideal of a multi-graded ring
- productOfProjectiveSpaces -- Constructs the multi-graded ring of a product of copies of P^1 (pp is a synonym)
- relativeEquations -- compute the relative quadrics
- relativeResolution -- compute the relative resolution
- relativeResolutionTwists -- compute the twists in the relative resolution
- resonanceDet -- compute the resonance determinant of the crucial constant strand of a degenerate K3 X_e(a,a)
- resonanceScroll -- compute the splitting type of the resonance scroll
- schemeInProduct -- multi-graded Ideal of the image of a map to a product of projective spaces
- schreyerName -- get the names of generators in the (nonminimal) Schreyer resolution according to Schreyer's convention
- smallDiagonal -- Ideal of the small diagonal in (P^1)^n

- Symbols
- FineGrading -- Option for carpet, canonicalCarpet
- Scrolls -- Option for carpet, canonicalCarpet