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Components with the expected codimension in the moduli scheme of stable spin curves

100%

Ballico E.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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2015

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tom 69

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nr 1

EN

Here we study the Brill–Noether theory of “extremal” Cornalba’s theta-characteristics on stable curves C of genus g, where “extremal” means that they are line bundles on a quasi-stable model of C with #(Sing(C)) exceptional components.

2

Gonality for stable curves and their maps with a smooth curve as their target

100%

Ballico E.

Open Mathematics

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2009

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tom 7

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nr 1

54-58

EN

Here we study the deformation theory of some maps f: X → ℙr , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of [...] with fixed topological type behaves like the stratification by gonality of M g.

3

On the real $X$-ranks of points of $\mathbb{P}^n(\mathbb{R})$ with respect to a real variety $X\subset\mathbb{P}^n$

100%

Ballico E.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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2010

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tom 64

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nr 2

EN

Let $X\subset\mathbb{P}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $\mathbb{R}$. For any $P \in \mathbb{P}^n(\mathbb{R})$ the real $X$-rank $r_{X,\mathbb{R}}(P)$ is the minimal cardinality of $S\subset X(\mathbb{R})$ such that $P\in \langle S\rangle$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.

4

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

100%

Ballico E.

Open Mathematics

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2005

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tom 3

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nr 2

183-187

EN

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(nn+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.

5

On projective degenerations of Veronese spaces

100%

Ballico E.

Banach Center Publications

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1996

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tom 37

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nr 1

45-51

EN

Here we give several examples of projective degenerations of subvarieties of $ℙ^{t}$. The more important case considered here is the d-ple Veronese embedding of $ℙ^{n}$; we will show how to degenerate it to the union of $d^{n}$ n-dimensional linear subspaces of $ℙ^{t}; t:= (n+d)/(n!d!) - 1$ and the union of scrolls. Other cases considered in this paper are essentially projective bundles over important varieties. The key tool for the degenerations is a general method due to Moishezon. We will give elsewhere several applications to postulation problems and to embedding problems.

6

Unique decomposition for a polynomial of low rank

64%

Ballico E., Bernardi A.

Annales Polonici Mathematici

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2013

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tom 108

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nr 3

219-224

EN

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of $ℙ^m$ into $ℙ^{{m+d \atop d}-1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F = M₁^d + ⋯ + M_t^d + Q$, where $M₁,. .., M_t$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q = ∑_{i=1}^q l_i^{d-d_i} m_i$ with $l_i$'s linear forms and $m_i$'s forms of degree $d_i$ such that $∑(d_i + 1) = s - t.$

7

On the existence of curves in $ℙ^n$ with stable normal bundle

64%

Ballico E., Ramella L.

Annales Polonici Mathematici

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1999

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tom 72

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nr 1

33-42

EN

We prove that for integers n,d,g such that n ≥ 4, g ≥ 2n and d ≥ 2g + 3n + 1, the general (smooth) curve C in $ℙ^n$ with degree d and genus g has a stable normal bundle $N_C$.

8

On the principle of real moduli flexibility: perfect parametrizations

64%

Ballico E., Ghiloni R.

Annales Polonici Mathematici

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2014

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tom 111

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nr 3

245-258

EN

Let V be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer b (arbitrarily large), there exists a trivial Nash family $𝒱 = {V_y}_{y ∈ R^b}$ of real algebraic manifolds such that V₀ = V, 𝒱 is an algebraic family of real algebraic manifolds over $y ∈ R^b∖{0}$ (possibly singular over y = 0) and 𝒱 is perfectly parametrized by $R^b$ in the sense that $V_y$ is birationally nonisomorphic to $V_z$ for every $y,z ∈ R^b$ with y ≠ z. A similar result continues to hold if V is a singular real algebraic set.

9

The principle of moduli flexibility for real algebraic manifolds

64%

Ballico E., Ghiloni R.

Annales Polonici Mathematici

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2013

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tom 109

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nr 1

1-28

EN

Given a real closed field R, we define a real algebraic manifold as an irreducible nonsingular algebraic subset of some Rⁿ. This paper deals with deformations of real algebraic manifolds. The main purpose is to prove rigorously the reasonableness of the following principle, which is in sharp contrast with the compact complex case: "The algebraic structure of every real algebraic manifold of positive dimension can be deformed by an arbitrarily large number of effective parameters".

10

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

64%

Ballico E., Malaspina F.

Open Mathematics

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2008

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tom 6

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nr 1

143-148

EN

Here we study vector bundles E on the Hirzebruch surface F e such that their twists by a spanned, but not ample, line bundle M = $$ \mathcal{O} $$ Fe(h + ef) have natural cohomology, i.e. h 0(F e, E(tM)) > 0 implies h 1(F e, E(tM)) = 0.

11

On Buchsbaum bundles on quadric hypersurfaces

52%

Ballico E., Malaspina F., Valabrega P., Valenzano M.

Open Mathematics

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2012

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tom 10

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nr 4

1361-1379

EN

Let E be an indecomposable rank two vector bundle on the projective space ℙn, n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q n ⊂ ℙn+1, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q 3, k ≥ 2, we prove two boundedness results.

12

Zero-dimensional subschemes of ruled varieties

52%

Ballico E., Bocci C., Fontanari C.

Open Mathematics

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2004

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tom 2

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nr 4

538-560

EN

Here we study zero-dimensional subschemes of ruled varieties, mainly Hirzebruch surfaces and rational normal scrolls, by applying the Horace method and the Terracini method

13

Smoothing of rational m-ropes

52%

Ballico E., Gasparim E., Köppe T.

Open Mathematics

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2009

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tom 7

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nr 4

623-628

EN

In a recent paper, Gallego, González and Purnaprajna showed that rational 3-ropes can be smoothed. We generalise their proof, and obtain smoothability of rational m-ropes for m ≥ 3.

Ograniczanie wyników

Components with the expected codimension in the moduli scheme of stable spin curves

Gonality for stable curves and their maps with a smooth curve as their target

On the real \(X\)-ranks of points of \(\mathbb{P}^n(\mathbb{R})\) with respect to a real variety \(X\subset\mathbb{P}^n\)

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

On projective degenerations of Veronese spaces

Unique decomposition for a polynomial of low rank

On the existence of curves in $ℙ^n$ with stable normal bundle

On the principle of real moduli flexibility: perfect parametrizations

The principle of moduli flexibility for real algebraic manifolds

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

On Buchsbaum bundles on quadric hypersurfaces

Zero-dimensional subschemes of ruled varieties

Smoothing of rational m-ropes