We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all 𝔖6-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, we check that 𝔖6-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as 𝔖6-representations.
Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.
This paper has been started as a particular application of the method of resolutions via Grobner bases we suggested here. We introduce a notion of a shuffle algebra. A shuffle algebra is a Z+-graded vector space V=∪∞i=1 such that for any pair (i,j) there exists a collection of operations ∗σ:Vi⊗Vj→Vi+j numbered by (i,j)-shuffle permutations σ∈Si+j (i.e. σ preserves the order of the first i elements and the order of the last j elements) yielding the natural associativity conditions. Enumerative problems for monomial shuffle algebras are in one-to-one correspondence with the pattern avoidance problems for permutations. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations. Both results generalizes the classical results for associative algebras. The first homological result is a generalization of the Golod-Shafarevich theorem and the second one generalizes the theory of Anick chains. It seems that most of particular applications we discuss are known to specialists but the general method was definitely not known. We hope that it will simplify a lot of work in this area. It is not hard to see that shuffle algebras form an interesting class of binary shuffle operads and illustrates quite well the importance of the latter notion.