Fix Integers Research Articles

On the base locus of linear systems of general double points

Fix integers n ≥ 1, d ≥ 4 and x>0 such that (n+1)(x-1) +Bin(n+2, 2) ≤ Bin(n+d, n). Take a general S ⊂ Pn such that #S=x and let B denote the scheme-theoretic base locus of |I2s(d)|, where 2S is the union of the double points with S as their reduction. Then 2S is the union of the connected components of B containing at least one point of S. We prove this theorem proving that a general union of x-1 double points and one triple point has no higher cohomology in degree d.

The norm of linear extension operators for [formula omitted

Fix integers m≥2, n≥1. We prove the existence of a bounded linear extension operator for Cm−1,1(Rn) with operator norm at most exp⁡(γDk), where D:=(m+n−1n) is the number of multiindices of length n and order at most m−1, and γ,k>0 are absolute constants (independent of m,n,E). Upper bounds on the norm of this operator are relevant to basic questions about fitting a smooth function to data. Our results improve on a previous construction of extension operators of norm at most exp⁡(γDk2D). Along the way, we establish a finiteness theorem for Cm−1,1(Rn) with improved bounds on the involved constants.

Random walks avoiding their convex hull with a finite memory

Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d=2 and k=1, we obtain the limiting speed explicitly: it is 8∕(9π2).

Existence of semistable vector bundles with fixed determinants

Let R be an excellent Henselian discrete valuation ring with algebraically closed residue field k of any characteristic. Fix integers r,d with r≥2. Let XR be a regular fibred surface over Spec(R) with special fibre denoted Xk, a generalised tree-like curve of genus g≥2. Let LR be a line bundle on XR of degree d such that the degree of the restriction of LR on the rational components of Xk is a multiple of r. In this article we prove the existence of a rank r locally free sheaf on XR of determinant LR such that it is semistable on the fibres.

A pathological case of the C1 conjecture in mixed characteristic

AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

Gaps in the pairs (border rank, symmetric rank) for symmetric tensors

Fix integers m 2, s 5 and d 2s + 2. Here we describe the possible symmetric tensor ranks 2d+s 7 of all symmetric tensors (or homogeneous degree d polynomials) in m + 1 variables with border rank s.

An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables

Fix integers m≥5 and d≥3. Let f be a degree d homogeneous polynomial in m+1 variables. Here, we prove that f is the sum of at most d·⌈(m+dm)/(m+1)⌉d-powers of linear forms (of course, this inequality is nontrivial only if m≫d.)

Refining Castelnuovo-Halphen bounds

Fix integers r,d,s,π with r≥4, d≫s, r−1≤s≤2r−4, and π≥0. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus pa(C) of an integral projective curve C⊂ℙr of degree d, assuming that C is not contained in any surface of degree π. Next we discuss other types of bound for pa(C), involving conditions on the entire Hilbert polynomial of the integral surfaces on which C may lie.

The de Bruijn–Erdős theorem for hypergraphs

Fix integers n ≥ r ≥ 2. A clique partition of \({{[n] \choose r}}\) is a collection of proper subsets \({A_1, A_2, \ldots, A_t \subset [n]}\) such that \({\bigcup_i{A_i \choose r}}\) is a partition of \({{[n]\choose r}}\) . Let cp(n, r) denote the minimum size of a clique partition of \({{[n] \choose r}}\) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3,$${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$We conjecture cp(n, r) = (1 + o(1))n r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1 + o(1))n r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n r) of the r-sets of [n] are covered. We also give an absolute lower bound \({{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}\) when n = q 2 + q + r − 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

Tensor ranks and symmetric tensor ranks are the same for points with low symmetric tensor rank

Fix integers n ≥ 1, d ≥ 2. Let V be an (n + 1)-dimensional vector space over a field with characteristic zero. Fix a symmetric tensor \({T\in S^d(V)\subset V^{\otimes d}}\). Here we prove that the tensor rank of T is equal to its symmetric tensor rank if the latter is at most (d + 1)/2.

On the size of maximal antichains and the number of pairwise disjoint maximal chains

Fix integers n and k with n ≥ k ≥ 3 . Duffus and Sands proved that if P is a finite poset and n ≤ | C | ≤ n + ( n − k ) / ( k − 2 ) for every maximal chain in P , then P must contain k pairwise disjoint maximal antichains. They also constructed a family of examples to show that these inequalities are tight. These examples are two-dimensional which suggests that the dual statement may also hold. In this paper, we show that this is correct. Specifically, we show that if P is a finite poset and n ≤ | A | ≤ n + ( n − k ) / ( k − 2 ) for every maximal antichain in P , then P has k pairwise disjoint maximal chains. Our argument actually proves a somewhat stronger result, and we are able to show that an analogous result holds for antichains.

Extremal Problems for $t$-Partite and $t$-Colorable Hypergraphs

Fix integers $t \ge r \ge 2$ and an $r$-uniform hypergraph $F$. We prove that the maximum number of edges in a $t$-partite $r$-uniform hypergraph on $n$ vertices that contains no copy of $F$ is $c_{t, F}{n \choose r} +o(n^r)$, where $c_{t, F}$ can be determined by a finite computation. We explicitly define a sequence $F_1, F_2, \ldots$ of $r$-uniform hypergraphs, and prove that the maximum number of edges in a $t$-chromatic $r$-uniform hypergraph on $n$ vertices containing no copy of $F_i$ is $\alpha_{t,r,i}{n \choose r} +o(n^r)$, where $\alpha_{t,r,i}$ can be determined by a finite computation for each $i\ge 1$. In several cases, $\alpha_{t,r,i}$ is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.

Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Fix integersr,s1,…,slr,s_{1},\dots ,s_{l}such that1≤l≤r−11\leq l\leq r-1andsl≥r−l+1s_{l}\geq r-l+1, and letC(r;s1,…,sl)\mathcal {C}(r;s_{1},\dots ,s_{l})be the set of all integral, projective and nondegenerate curvesCCof degrees1s_{1}in the projective spacePr\mathbf {P}^{r}, such that, for alli=2,…,li=2,\dots ,l,CCdoes not lie on any integral, projective and nondegenerate variety of dimensioniiand degree>si>s_{i}. We say that a curveCCsatisfies theflag condition(r;s1,…,sl)(r;s_{1},\dots ,s_{l})ifCCbelongs toC(r;s1,…,sl)\mathcal {C}(r;s_{1},\dots ,s_{l}). DefineG(r;s1,…,sl)=max⁡{pa(C):C∈C(r;s1,…,sl)},G(r;s_{1},\dots ,s_{l})=\operatorname {max}\left \{p_{a}(C):\,C\in \mathcal {C}(r;s_{1},\dots ,s_{l})\right \},wherepa(C)p_{a}(C)denotes the arithmetic genus ofCC. In the present paper, under the hypothesiss1≫⋯≫sls_{1}\gg \dots \gg s_{l}, we prove that a curveCCsatisfying the flag condition(r;s1,…,sl)(r;s_{1},\dots ,s_{l})and of maximal arithmetic genuspa(C)=G(r;s1,…,sl)p_{a}(C)=G(r;s_{1},\dots ,s_{l})must lie on a unique flag such asC=Vs11⊂Vs22⊂⋯⊂Vsll⊂PrC=V_{s_{1}}^{1}\subset V_{s_{2}}^{2}\subset \dots \subset V_{s_{l}}^{l}\subset {\mathbf {P}^{r}}, where, for anyi=1,…,li=1,\dots ,l,VsiiV_{s_{i}}^{i}denotes an integral projective subvariety ofPr{\mathbf {P}^{r}}of degreesis_{i}and dimensionii, such that its general linear curve section satisfies the flag condition(r−i+1;si,…,sl)(r-i+1;s_{i},\dots ,s_{l})and has maximal arithmetic genusG(r−i+1;si,…,sl)G(r-i+1;s_{i},\dots ,s_{l}). This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.

An intersection theorem for four sets

Fix integers n , r ⩾ 4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A , B , C , D ∈ F with | A ∪ B ∪ C ∪ D | ⩽ 2 r , we have A ∩ B ∩ C ∩ D ≠ ∅ . We prove that for n sufficiently large, | F | ⩽ ( n − 1 r − 1 ) , with equality only if ⋂ F ∈ F F ≠ ∅ . This is closely related to a problem of Katona and a result of Frankl and Füredi [P. Frankl, Z. Füredi, A new generalization of the Erdős–Ko–Rado theorem, Combinatorica 3 (3–4) (1983) 341–349], who proved a similar statement for three sets. It has been conjectured by the author [D. Mubayi, Erdős–Ko–Rado for three sets, J. Combin. Theory Ser. A, 113 (3) (2006) 547–550] that the same result holds for d sets (instead of just four), where d ⩽ r , and for all n ⩾ d r / ( d − 1 ) . This exact result is obtained by first proving a stability result, namely that if | F | is close to ( n − 1 r − 1 ) then F is close to satisfying ⋂ F ∈ F F ≠ ∅ . The stability theorem is analogous to, and motivated by the fundamental result of Erdős and Simonovits for graphs.

Surjections of vector bundles and cohomology classes

Fix integers r>s> 0. Let E be a rank r vector bundle on a smooth n-dimensional variety, n ≥ 2. Here we discuss the existence of a torsion free rank s sheaf F on X and a surjection f : E → F such that the induced maps f∗,i : H i (X, E) → H i (X, F) are bijective for all 0 ≤ i ≤ n − 1.

Apolar Schemes of Algebraic Forms

AbstractThis is a note on the classical Waring's problem for algebraic forms. Fix integers (n, d, r, s), and let ∧ be a general r-dimensional subspace of degree d homogeneous polynomials in n+1 variables. Let denote the variety of s-sided polar polyhedra of ∧. We carry out a case-by-case study of the structure of for several specific values of (n, d, r, s). In the first batch of examples, is shown to be a rational variety. In the second batch, is a finite set of which we calculate the cardinality.

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( n n+x ). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension ( n /n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S.

Projective varieties with projectively equivalent singular 0-dimensional sections

Fix integers m, n such that 1 ≤ m ≤ n − 3. Let X ⊂ Pn be an integral non-degenerate m-dimensional variety. Assume either char(K) = 0 or char(K) > deg(X). Here we prove that all general 0-dimensional sections of X containing a tangent vector to a smooth point of X are protectively equivalent if and only if n − m + 1 ≤ deg(X) ≤ n − m + 2.

Curves in Grassmannians and Plücker Embeddings: The Injectivity Range

Fix integers g, r, d, v with g ≥ 2, r ≥ 2 and r < v ≤ d + r – rg. Let X be a smooth curve of genus g and E a general stable vector bundle on X with rank (E) = r and deg (E) = d. Assume E spanned and take a general linear subspace V of H0 (X, E) with dim (V) = v. Let hV : X → G(r, V) be the induced map into a Grassmannian. Here we give conditions on d and v which assure that the natural map (X, det (E)) is injective. Use the Plücker embedding to see G(r, V) as a subvariety of P. According to M. Teixidor i Bigas this means that hV(X) ⊂ P spans P

New Lower Bounds for Ramsey Numbers of Graphs and Hypergraphs

Let G be an r-uniform hypergraph. The multicolor Ramsey number rk(G) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K(r)n yields a monochromatic copy of G. Improving slightly upon results from M. Axenovich, Z. Füredi, and D. Mubayi (J. Combin. Theory Ser. B79 (2000), 66–86), we prove thattk2+1≤rk(K2,t+1)≤tk2+k+2,where the lower bound holds when t and k are both powers of a prime p. When t=1, we improve the lower bound by 1, proving that rk(C4)≥k2+2 for any prime power k. This extends the result of F. Lazebnik and A. J. Woldar (J. Combin. Theory Ser. B79 (2000), 172–176), which proves the same bound when k is an odd prime power. These results are generalized to hypergraphs in the following sense. Fix integers r,s,t≥2. Let H(r)(s,t) be the complete r-partite r-graph with r−2 parts of size 1, one part of size s, and one part of size t (note that H(2)(s,t)=Ks,t). We prove thattk2−k+1≤rk(H(r)(2,t+1))≤tk2+k+r,where the lower bound holds when t and k are both powers of a prime p and(1−o(1))ks≤rk(H(r)(s,t))≤O(ks),for fixedt,s≥2,t>(s−1)!,rk(H(r)(3,3))=(1+o(1))k3.Some of our lower bounds are special cases of a family of more general hypergraph constructions obtained by algebraic methods. We describe these, thereby extending results of F. Lazebnik and A. J. Woldar (J. Graph Theory, 2001) about graphs.