Wahl maps and extensions of canonical curves and K⁢3K3 surfaces

Let C be a smooth projective curve (resp. {(S,L)} a polarized {K3} surface) of genus {g\geqslant 11}, with Clifford index at least 3, considered in its canonical embedding in {\mathbb{P}^{g-1}} (resp. in its embedding in {|L|^{\vee}\cong\mathbb{P}^{g}}). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in {\mathbb{P}^{g+r}}, not a cone, with {\dim(Y)=r+2} and {\omega_{Y}=\mathcal{O}_{Y}(-r)}, if the cokernel of the Gauss–Wahl map of C (resp. {\operatorname{H}^{1}(T_{S}\otimes L^{\vee})}) has dimension larger than or equal to {r+1} (resp. r). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

$${\mathfrak {S}}_5$$-equivariant syzygies for the Del Pezzo surface of degree 5

The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$P5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$S5. We give canonical explicit $${\mathfrak {S}}_5$$S5-invariant Pfaffian equations through a 6$$\times $$×6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$S5. Finally, we give $${\mathfrak {S}}_5$$S5-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$(P1)5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

Glicci ideals

A central problem in liaison theory is to decide whether every arithmetically Cohen–Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can indeed be achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+ 1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen–Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of simple points in finitely many steps, and hence to a complete intersection in projective 4-space.

On Complete Intersections Contained in Cohen-Macaulay and Gorenstein Ideals

We look for complete intersections containing certain arithmetically Cohen-Macaulay schemes, and give a complete description in the case of 2-codimensional arithmetically Cohen-Macaulay schemes and 3-codimensional arithmetically Gorenstein schemes. In particular, we prove that in these cases the sets of types of complete intersections containing such schemes have a unique minimal element and we compute it.