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Let X ⊂ P r be an integral and non-degenerate variety. For any q ∈ P r its X-rank r X (q) is the minimal cardinality of a finite subset of X whose linear span contains q. The solution set S (X , q) of q ∈ P r is the set of all S ⊂ X such that # S = r X (q) and S spans q. We prove that if X ≠ P r there is at least one q with # S (X , q) > 1 and that for almost all pairs (X, q) we have dim S (X , q) > 0 . [ABSTRACT FROM AUTHOR]

Let F⊂ P2× P2v be the 3-dimensional flag. Let π1 F→ P2 and π2 F→ P2v be the projections. For all u,v ∈N\{(0,0)} let M(u,v) denote the set of all curves π1-1(F) ∪ π2-1(E) such that π1-1(F) ∩ π2-1(E)=∅, #F=v and #E=u. Any A∈ M(u,v) has u+v connected components, all of them smooth and rational and embedded as lines by the Segre embedding of F⊂ P2× P2v. In this paper we study the bigraded Hilbert function H0(IA(a,b)), (a,b)∈N2, for a general A∈M(u,v). We also give geometric properties of IA(a,b) (spannedness and a uniqueness result for non-general A∈ M(u,v)). [ABSTRACT FROM AUTHOR]

Identifiability holds for the k-secant variety σk(X) of an embedded variety X ⊂ P r if a general q ∈ σk(X) is in the linear span of a unique subset of X with cardinality k. Identifiability is true if the general tangential k-contact locus Γk ⊂ X has dimension 0. Under certain assumptions on dim ⁡ σ x (X) for some specific x>k, we get dim ⁡ Γ k = 0. Here we give some conditions which exclude the case Γx a hypersurface for some x>k and get dim ⁡ Γ k = 0 and hence identifiability for the k-secant variety. As an example we consider the case of Segre-Veronese embeddings of multiprojective spaces, in which the elements of P r corresponds to partially symmetric tensors. [ABSTRACT FROM AUTHOR]

We give examples of infinitely extendable (not as cones) arithmetically Cohen–Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension 2 subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension 3 subschemes due to J. O. Kleppe and R.-M. Miró-Roig. [ABSTRACT FROM AUTHOR]

Let q be a prime power, q ≠ 2 , 3 , and let A be an n × n matrix over q 2 . We prove that the numerical range of A B and B A is the same for every rank 1 matrix B if and only if A is a multiple of a unitary matrix. The corresponding result for complex matrices was proved by Chien, Gau, Li, Tsai and Wang [Products of operators and numerical range, Linear Multilinear Algebra 64(1) (2016) 58–67]. [ABSTRACT FROM AUTHOR]

We define linear codes which are s-Locally Recoverable Codes (or s-LRC), i.e. codes which are LRC in s ways, the case s = 1 roughly corresponding to the classical case of LRC codes. We use them to describe codes which correct many erasures, although they have small minimum distance. Any letter of a received word may be corrected using s different local codes. We use the Segre embedding of s local codes and then a linear projection. [ABSTRACT FROM AUTHOR]

We study the linear algebra of finite subsets S of a Segre variety X. In particular we classify the pairs (S, X) with S linear dependent and # (S) ≤ 5 . We consider an additional condition for linear dependent sets, i.e. that no two of their points are contained in a line of X, and get far better lower bounds for # (S) in term of the dimension and number of the factors of X. In this discussion and in the classification of the case # (S) = 5 , X ≅ P 1 × P 1 × P 1 we use the rational normal curves contained in X. [ABSTRACT FROM AUTHOR]

We prove the existence of many homogeneous polynomials with non-unique minimal additive decompositions. For all integers n, d, k such that n ≥ 2 , d ≥ 6 and ⌈ d / 2 ⌉ < k ≤ n + ⌊ d / 2 ⌋ n there is a degree d form in n + 1 variables with rank k and having infinitely many decompositions as the sum of kd-powers of linear forms. We also construct different families of forms f and for each of them we compute the dimension (or the number) of its additive decompositions. [ABSTRACT FROM AUTHOR]

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees. [ABSTRACT FROM AUTHOR]

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a byproduct, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on multiple hyperplanes and prove the existence of such bundles that do not split if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most 3/2 on the double plane and describe the family of isomorphism classes of them. [ABSTRACT FROM AUTHOR]

We consider the existence of space curves with prescribed degree d and genus g and not contained in a surface of prescribed degree m - 1 when g is below the maximal genus allowed by the pair (d, m), mainly in the so-called Range A and in the part d ≤ m (m + 1) / 2 of Range B. [ABSTRACT FROM AUTHOR]

We give quantitative and qualitative results on the family of surfaces in CP 3 containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterward we prove that twistor lines are Zariski dense in the Grassmannian Gr(2, 4). Then, for any degree d ≥ 4 , we give lower bounds on the maximum number of twistor lines contained in a degree d surface. The smooth and singular cases are studied as well as the j-invariant one. [ABSTRACT FROM AUTHOR]

We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification. [ABSTRACT FROM AUTHOR]

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967-980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities. [ABSTRACT FROM AUTHOR]

We study the partially symmetric tensorsTwith low rank whose rank is not achieved by a unique set when the rank has an upper bound like, whereis the minimum of the entries of the multidegreeofT. For the proof, we use the Hilbert function of finite subsets of a product of projective spaces with respect to a Segre–Veronese embedding. [ABSTRACT FROM AUTHOR]

Let X⊂Pr be an integral and non-degenerate variety non-singular in codimension 1. Let τ(X,2) = τ(X)⊂Pr be the tangential variety of X. Set n := dim(X). For any integer b > 2 let τ(X,b) be the join of τ(X) and b - 2 copies of X. Mimicking the notion of weakly κ-degenerate varieties in-troduced by Chiantini and Ciliberto we give a conditions on X assuring that a general q ∊ r(X,b) is in the linear span of a unique scheme Z⊂X with Z = v U {p1,... ,Pb-2} with v connected of degree 2 and v ⊂ Xreg. [ABSTRACT FROM AUTHOR]

Let X be a smooth n-fold, $$n\ge 3$$ , and $$\mathcal {L}$$ a spanned and ample line bundle on X with $$h^1(\mathcal {O}_X)=0$$ . Let $$\mathcal {E}$$ be a spanned rank r vector bundle on X with $$\det (\mathcal {E})\cong \mathcal {L}$$ and no trivial factor. Here we prove that $$r\le h^0(\mathcal {L})-1$$ and prove the existence of some $$\mathcal {E}$$ for all r with $$n\le r \le h^0(\mathcal {L})-1$$ . In the case $$n=3$$ we show that if $$h^0(\mathcal {L}) -r$$ is very small (in terms of the t-connectedness of $$(X,\mathcal {L})$$ ), then $$\mathcal {E}$$ is obtained by a standard procedure from $$\mathcal {L}$$ . If $$h^1(\mathcal {O}_X)>0$$ we prove the upper bound for r substituting the integer $$h^0(\mathcal {L})-2$$ with an integer $$\rho (\mathcal {L})$$ with $$h^0(\mathcal {L})-2+ \max \{0,2h^1(\mathcal {O}_X)-h^1(\mathcal {L})\}\le \rho (\mathcal {L})\le h^0(\mathcal {L})-2+2h^1(\mathcal {O}_X)$$ . [ABSTRACT FROM AUTHOR]

LetC ⊂ ℙnbe a smooth curve andNCits normal bundle.NCsatisfies strong interpolation if for all integerss > 0 andλi ∈ {0,1,…,n−1}, 1 ≤ i ≤ s, there are distinct pointsand linear subspacessuch thatfor alliand a certain maphas maximal rank (Atanasov). We prove thatNCsatisfies strong interpolation if eitherCis a linearly normal elliptic curve orCis a general embedding of degreed ≥ (5n−8)g + 2n2−5n + 4 of a smooth curveXof genusg ≥ 2. [ABSTRACT FROM PUBLISHER]

Let X be a smooth curve of genus g defined over $$\mathbb {F}_q$$ . In this note we study the maximal number of points $$P_1,\ldots ,P_n\in X(\mathbb {F}_q)$$ such that $$h^0(\mathcal {O}_X(a_1P_1+\cdots +a_nP_n)) =1$$ for all $$(a_1,\ldots ,a_n)\in {\mathbb {N}_0} ^n$$ with $$a_1+\cdots +a_n \le g$$ . We also look at the Weierstrass n-semigroup of n points of $$X(\mathbb {F}_q)$$ when $$n>q$$ . [ABSTRACT FROM AUTHOR]

We study the postulation of a general union $${X\subset \mathbb{P}^3}$$ of one multiple point mP and t disjoint lines. We prove that it has the expected Hilbert function, proving a conjecture by Carlini, Catalisano and Geramita. [ABSTRACT FROM AUTHOR]

We study the gonality and the existence of low degree pencils on curves with a model on a weighted projective plane, when their singu- larities are only ordinary nodes or ordinary cusps and they are general in the weighted projective plane. [ABSTRACT FROM AUTHOR]

We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to prove a Torelli-type theorem in some cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of determining the Hilbert function of a general union $${X\subset \mathbb{P}^n}$$ of an m-point mP and a general smooth rational curve of degree s. If $${n\geq 4}$$ , then X has the expected Hilbert function, i.e., X has maximal rank. If n = 3, then there are exceptional cases (if $${3\leq s\leq m}$$ ). We conjecture that these are the only exceptional cases and we prove this conjecture in several cases (e.g., if $${s\gg m}$$ ). [ABSTRACT FROM AUTHOR]

We study the decomposition of a degree d homogeneous linear form ƒ ∈ K[x0,⋅⋅⋅,xn] as a sum of x d-powers of linear forms in x0,⋅⋅⋅,xn and y d-powers of x1,⋅⋅⋅,xn. Most of our results are for cases with ny ≤ (n+d-1n-1) and with d ≥ 5 . If n ≤ 4, we also consider the cases d= 3,4. We also study other interpolation problems (e.g., flags instead of a hyperplane). [ABSTRACT FROM AUTHOR]

Let X be a smooth curve of genus g > 0. For any $${n \geq 2}$$ and any n distinct points $${P_{1}, . . . ,P_{n} \in X, \,\,{\rm let}\,\, H(P_1, . . . ,P_n)}$$ be the set of all $${(a_{1}, . . . ,a_{n}) \in \mathbb{N}^n {\rm such}\,{\rm that}\,\, \mathcal{O}_{X}(a_{1}P_{1} + \cdots + a_{n}P_{n})}$$ is spanned. We raise some questions on the minimal number of generators of the semigroup $${H(P_{1}, . . . ,P_{n})}$$ and test it in a few cases. [ABSTRACT FROM AUTHOR]

Fix a real tensorof format. Here we prove that its real tensor rank is at mosttimes its complex tensor rank. A similar bound is true for symmetric tensors and for any perfect field (e.g. a finite field). [ABSTRACT FROM PUBLISHER]

Let C be a smooth curve of genus g. For each positive integer r the birational r-gonality sr(C) of C is the minimal integer t such that there is L ∈ Pict(C) with h0(C,L) = r + 1. Fix an integer r ≥ 3. In this paper we prove the existence of an integer gr such that for every integer g ≥ gr there is a smooth curve C of genus g with sr+1(C)/(r + 1) > sr(C)/r, i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails [ABSTRACT FROM AUTHOR]

We investigate the reflexive sheaves on $${\mathbb{P}^3}$$ spanned in codimension 2 with very low first Chern class c. We also give the sufficient and necessary conditions on numeric data of such sheaves for indecomposabiity. As a by-product we obtain that every reflexive sheaf on $${\mathbb{P}^3}$$ spanned in codimension 2 with c = 2 is spanned. [ABSTRACT FROM AUTHOR]

We prove that a general union X ⊂ ℙ³ P of prescribed numbers of lines and double points hasmaximal rank, except a few well-known exceptional cases. [ABSTRACT FROM AUTHOR]

We prove that a general union X - P³ of prescribed numbers of lines and double points has maximal rank, except a few well-known exceptional cases. [ABSTRACT FROM AUTHOR]

We study the surjectivity of suitable weighted Gaussian maps $$\gamma _{a,b}(X,L)$$ which provide a natural generalization of the standard Gaussian maps and encode the local geometry of the locus $$\mathfrak{Th }^r_{g,h}\subset \mathcal M _g$$ of curves endowed with an $$h$$ -th root $$L$$ of the canonical bundle satisfying $$h^0(L)\ge r+1$$ . In particular, we get a bound on the dimension of its Zariski tangent space, hich turns out to be sharp in the special case $$r=0$$ . Finally, we describe this locus in the case of complete intersection curves. [ABSTRACT FROM AUTHOR]

Here we study the small weight codewords and the generalized Hamming weights of Goppa codes on a smooth plane curve C in terms of the intersection of C with lines and conics. [ABSTRACT FROM AUTHOR]

In this article, we construct, for everyn, smooth varieties of general type of dimensionnwith the firstplurigenera equal to zero. Hacon-McKernan, Takayama, and Tsuji have recently shown that there are numbersrnsuch that ∀ r ≥ rn, ther − canonical map of every variety of general type of dimensionnis birational. Our examples show thatrngrows at least quadratically as a function ofn. Moreover, they show that the minimal volume of a variety of general type of dimensionnis smaller than. In addition, we prove that for every positive rational numberqthere are smooth varieties of general type with volumeqand dimension arbitrarily big. [ABSTRACT FROM PUBLISHER]

Let C ⊂ Pn+1 be a rational normal curve and let X ⊂ Pn be one of its tangential projection. We describe the X-rank of a point P ϵPn in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P. [ABSTRACT FROM AUTHOR]

We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties. [ABSTRACT FROM AUTHOR]

Here we improve the lower bound on the minimum distance for the evaluation codes obtained by Davis from certain blowing-ups of the planes (anticanonical surfaces). [ABSTRACT FROM AUTHOR]

Let vd : ℙm → ℙn, n : = (...) -1, denote the degree d Veronese embedding of ℙm. For any P ∈ ℙn, lets sr(P) be the minimal cardinality of ⊂ vd(ℙm) such that P ∈ (S). Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of #(S) powers Ld with L a linear form (or a sum of #(S) d-powers of vectors). Here we fix an integral variety T ⊋ ℙm and P ∈ d (T)> and study a similar decomposition with S ⊋ T and # (S) minimal. For instance, if T is a linear subspace, then we prove that #(S) ≥ #(S ∩ T) + d + 1 and classify all (S, P) such that #(S) - #(S #8745; T) ≤ 2d - 1. [ABSTRACT FROM AUTHOR]

We study the Hilbert function of the union X of a given zero-dimensional scheme Z ₎ ℙr, r ≥ 4, and t general lines of ℙr, proving, for example, that if t » deg(Z), then X has maximal rank. [ABSTRACT FROM AUTHOR]

Abstract. Fix integers d; g; r such that r ≥ 3, g > 0 and d ≥ g + r. Here we prove the existence of an integral non-special curve C in an r-dimensional projective space such that deg(C) = d, pa (C) = g, C has exactly one node and C has maximal rank (i.e. it has the expected postulation), i.e., the general non-special embedding of a general curve with a single node has maximal rank. [ABSTRACT FROM AUTHOR]

Here we study the existence of scroll codes with certain parameters (following papers by T. Johnsen, N. H. Rasmussen and T. Nakashima). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Here we study the postulation of general disjoint unions of planes and lines in ℙ using the Horace Method and some unreduced structures called here 2-sundials. [ABSTRACT FROM AUTHOR]

Here we study curves C over $${\mathbb{F}_q}$$ contained in a hyperbolic quadric surface and such that $${\sharp (C(\mathbb{F}_q))}$$ is large. [ABSTRACT FROM AUTHOR]

lgebraic varieties of projective spaces over the algebraic closure of a finite field, with two or few characters with respect to r-dimensional linear spaces are studied. [ABSTRACT FROM AUTHOR]

We study several interpolation problems for homogeneous polynomials over a finite field. [ABSTRACT FROM AUTHOR]

Here we give sufficient conditions for the existence of an α-stable (for all α > 0) coherent system (i.e. a vector bundle together with a linear subspace of its global sections) of type (d,n,k), k > n, on a smooth projective curve. [ABSTRACT FROM AUTHOR]

Let ( E ,????) be a complex normed space, E ' its dual and B 1 ??? E ' the dual of the unit ball of E . Equip E ' and hence B 1 with the weak*-topology s( E ,? E '), but call E ' k the space E ' with the kelleyfication topology of s( E ,? E ') and use the corresponding complex structure. Here we prove that for every q ?=?2 and for every i ?=?1. Let M be a complex manifold locally modelled over an infinite-dimensional Banach space with countable unconditional basis and with the localizing property. Let S ??? M be a discrete subset and E a locally free sheaf on M with finite rank. Here we prove that the natural map is bijective. [ABSTRACT FROM AUTHOR]

Let X be a topological space whose topology may be defined by a complete metric. Douady (case X compact) and Pestov (the general case) proved that X may be equipped with a structure of Banach analytic set. Here we prove that if X is not discrete, then this may be done in uncountably many ways. Let ( Y ,? d ) be a non-complete metric space and its completion. Here we prove that we may see as a closed analytic subset of a Banach space in such a way that with V a dense linear subspace of . Then we will prove (using bounded holomorphic functions) the following result. Let X be a connected open subset of an infinite-dimensional locally convex topological vector space V with the weak topology and A an open domain of such that has positive capacity. Then X ?×? A is not biholomorphic to X . [ABSTRACT FROM AUTHOR]