We study the minimal free resolution of the Veronese modules, S n,d,k = ⊕ i≥0 S k+id , where S = K[x 1 , . . . , x n ], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that S n,d,k is Cohen-Macaulay if and only if k < d, and that its minimal resolution is pure and has some linearity features when k > d(n − 1) − n. We prove combinatorially that the resolution of S 2,d,k is pure. We show that HS(S n,d,k ; z) =As an application, we calculate the complete Betti diagrams of the Veronese rings K[x, y, z] (d) , for d = 4, 5, and K[x, y, z, u] (3) .Given a graded ring S = ⊕ i≥0 S i , the Veronese subring S (d) is defined as ⊕ i≥0 S id and the Veronese modules S n,d,k , which are modules over the Veronese subring, are ⊕ i≥0 S k+id . In this paper, we set S = K[x 1 , . . . , x n ], where K is a field, and we deal with the syzygies of the Veronese modules. Since S (d) can be presented as R/I, where R is a polynomial ring and I a binomial ideal, in the following, we will consider S n,d,k as an R-module (see Section 1).There has been a lot of effort already to find the graded Betti numbers of the Veronese ring S (d) . The problem can be really hard, namely, in [10], Ein and Lazarsfeld showed that for d ≫ 0 the graded Betti numbers β i,j (S (d) ) = 0 for many j if i is large: in particular, they proved that, for d ≫ 0 there exist l 1 , l 2 such that β p,p+q = 0 for all p in the range l 1 d q−1 ≤ p ≤ d+n n − l 2 d n−q . It is known (see references in [17]) that β i = β i,(i+1)d , for all i > 0, in the cases n = 2 or (d, n) = (2, 3). Instead, when d = 2 and n > 3, we have that the equality holds for i ≤ 5. In addition, for d = 2, all Betti numbers have been determined. In case n, d ≥ 3, Ottaviani and Paoletti (in [17])
In 2009 Ekedahl introduced certain cohomological invariants of finite groups which are naturally related to the Noether Problem. We show that these invariants are trivial for every finite group in GL3 (k) and for the fifth discrete Heisenberg group H5. Moreover in the case of finite linear groups with abelian projective reduction, these invariants satisfy a recurrence relation in a certain Grothendieck group for abelian groups.Let V be a finite dimension faithful linear representation of a finite group G over a field k of characteristic prime to the order of G. holds in the Kontsevich value ring K 0 (Var k ) of algebraic k-varieties.All the known cases, where this equality fails are counterexamples to the Noether Problem. In the beginning of the last century, Noether [13] wondered about the rationality of the field extension k(V ) G /k for any finite group G and any field k, where k(V ) G are the invariants of the field of rational functions k(V ) over the regular representation V of G. (The Noether Problem can be stated for any arbitrary field, but we will not need the full generality.) The first counterexample, Q(V ) Z /47Z /Q, was given by Swan in [18] and it appeared during 1969. In the 1980s more counterexamples were found: for every prime p Saltman [16] and Bogomolov [3] showed that there exists a group of order p 9 and, respectively, of order p 6 such that the extension C(V ) G /C is not rational. Saltman used the second unramified cohomology group of the field, as a cohomological obstruction to rationality. Later, Bogomolov found a group cohomology expression for H 2 nr (C(V ) G , Q /Z) which now bears his name and is denoted by B 0 (G). Recently Hoshi, Kang and Kunyavskii investigateed the case where |G| = p 5 . They showed that B 0 (G) = 0 if and only if G belongs to the isoclinism family φ 10 ; see [8].In 2009, Ekedahl [5] defines, for every integer k, a cohomological mapwhere L 0 (Ab) is the group generated by the isomorphism classes {G} of finitely generated abelian groups G under the relation {A ⊕ B} = {A} + {B}. Let L i be the class of the affine space} for every smooth and proper k-variety X (for more details see Section 3 in [11]).The class {B G} of the classifying stack of G is an element of K 0 (Var k ) (see Proposition 2.5.b in [11]) and so one can define: Definition 1.2. For every integer i, the i-th Ekedahl invariant e i (G) of the group G is H −i ({B G}) in L 0 (Ab). We say that the Ekedahl invariants of G are trivial if e i (G) = 0 for all integer i = 0.In Proposition 2.5.a of [11], the author rephrases the equality (1) in terms of algebraic stacks, using the expressionSince {GL(V )} is invertible in K 0 (Var k ), the equation (1) ]). Assume char(k) = 0. If G is a finite group, then e i (G) = 0 for every i < 0, e 0 (G) = {Z}, e 1 (G) = 0 and e 2 (G) = {B 0 (G) ∨ }, where B 0 (G) ∨ is the pontryagin dual of the Bogomolov multiplier of the group G. Moreover, for i > 0, the invariant e i (G) is an integer linear combination of classes of finite abelian groups.Using that e 2 (G) = {B...
We study the intersection lattice of the arrangement A G of subspaces fixed by subgroups of a finite linear group G. When G is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of G. We generalize the notion of finite reflection groups. We say that a group G is generated (resp. strictly generated) in codimension k if it is generated by its elements that fix point-wise a subspace of codimension at most k (resp. precisely k).If G is generated in codimension two, we show that the intersection lattice of A G is atomic. We prove that the alternating subgroup Alt(W ) of a reflection group W is strictly generated in codimension two; moreover, the subspace arrangement of Alt(W ) is the truncation at rank two of the reflection arrangement A W .Further, we compute the intersection lattice of all finite subgroups of GL 3 (R), and moreover, we emphasize the groups that are "minimally generated in real codimension two", i.e, groups that are strictly generated in codimension two but have no real reflection representations. We also provide several examples of groups generated in higher codimension.
In this work, we define cooperative games on simplicial complexes, generalizing the study of probabilistic values of Weber [Web88] and quasi-probabilistic values of Bilbao, Driessen, Jiménez Losada and Lebrón [BDJLL01]. Applications to Multi-Touch Attribution and the interpretability of the Machine-Learning prediction models motivate these new developments [LL17, RSG16, SK13, SPK17, DSZ16, BBM + 15].We deal with the axiomatization provided by the λ i -dummy and the monotonicity requirements together with a probabilistic form of the symmetric and the efficiency axioms. We also characterize combinatorially the set of probabilistic participation influences as the facet polytope of the simplicial complex.A cooperative game is a pair (n, v) where n is a positive integer and v is the worth function v : 2 n → R, where 2 n is the power set of [n] def = {1, . . . , n}. We assume that v(∅) = 0. The elements of [n] are players of the game that may join in coalitions; a coalition T is a subset of [n] and v(T ) is the number of payoff of T in the cooperative game. For every player i an individual value φ i (v) is a (linear) function measuring the additional worth that i provides to a coalition during the cooperative game (n, v). The study of such values was extremely relevant for the community and we would like to highlight here a few important works of Shapley [Sha53,Sha72] and Weber [Web88] that have influenced the author.Recently, quite a lot of effort has been done to study cooperative games on matroids [BDJLL01, BDJLL02, MTMZ19, MZ11, FV11, NZKI97, Zha99]. Inspired by this recent articles, in this manuscript we define cooperative games on simplicial complexes and we study quasi-probabilistic values for such games.
In this paper, we sharpen the lower bound on the codimension of the irreducible components of the Noether–Lefschetz locus of surfaces in projective toric threefolds given in [U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1–22]. We also provide a simpler proof of Theorem 4.11 in [U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1–22], which allows one to avoid some technical assumptions.
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