The degree of a projective subscheme has an upper bound in term the codimension and reduction number. If variety almost maximal degree, that is, equals to minus one, then its Betti table been described explicitly. We build on this work by showing for most such varieties, defining ideals are componentwise linear particular linearity is suitable classifying tables varieties. As application, we co...

Let ` be a prime number and let k be an algebraically closed field with char(k) 6= `. Let K/k be a function field of transcendence degree td(K/k) > 1 with absolute Galois group GK . Take G K = GK and for all i ≥ 1, let G (i+1) K = [G , G](G) ∞ . In other words, the groups {G K }i≥1 form the descending central `∞-series of GK . Then ΠK = G /G = Gal(K ′/K) is the Galois group of the maximal pro-`...

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [33], [34], we give syntactical conditions necessary and sufficient for a relation to be intrinsically Π1 on a structure. We consider some examples of computable structures A and intrinsically Π1 relations R. We also consider a general family o...