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R. M. Miró-Roig

Publications and source records attributed to R. M. Miró-Roig.

14 recordsLinked to original sources
arXiv2020

Waring and cactus ranks and Strong Lefschetz Property for annihilators of symmetric forms

In this note we show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the Strong Lefschetz Property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the Strong Lefschetz Property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

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arXiv2019

Theta divisors and Ulrich bundles on Geometrically ruled surfaces

We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.

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arXiv2015

Homogeneous Ulrich bundles on Flag manifolds

Let $V$ be a $K$-vector space of dimension $n+1$. In this paper, we focus our attention into the existence of irreducible homogeneous Ulrich bundles on flag manifolds $\FF(p, q,n)$ which parameterizes all chains of linear subspaces $L_{p} \subset L_{q} \subset \PP(V)$ of dimension $p< q$, respectively. We determine all irreducible homogeneous Ulrich bundles on $\FF(0,n-1,n)$ and we prove that there are exactly $2^{n-1}$. Similarly, we prove that $\FF(0,n-2,n)$ and $\FF(1,n-1,n)$ are also the support of irreducible homogeneous Ulrich bundles. On the other hand, we prove that $\FF(0,1,n)$ do not support any irreducible homogeneous Ulrich bundle. We end posing a conjecture concerning the existence of irreducible homogeneous Ulrich bundles on $\FF(p,q,n)$ in terms of $p$ and $q$.

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arXiv2014

On the intersection of ACM curves in $\PP^3$

Bezout's theorem gives us the degree of intersection of two properly intersecting projective varieties. As two curves in P^3 never intersect properly, Bezout's theorem cannot be directly used to bound the number of intersection points of such curves. In this work, we bound the maximum number of intersection points of two integral ACM curves in P^3. The bound that we give is in many cases optimal as a function of only the degrees and the initial degrees of the curves.

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arXiv2014

Rational families of instanton bundles on $P^{2n+1}$

This paper is devoted to the theory of symplectic instanton bundles on an odd dimensional projective space ${\mathbb P}^{2n+1}$ with $n\ge 2$. We study the 't Hooft instanton bundles introduced by Ottaviani and a new family of instanton bundles which generalizes one introduced on ${\mathbb P}^3$ independently by Rao and Skiti. The main result is the determination of the birational types of the moduli spaces of 't Hooft and of Rao-Skiti instanton bundles, respectively. Assuming a conjecture of Ottaviani, we show that the moduli space of all symplectic instanton bundles on ${\mathbb P}^{2n+1}$ with $n\ge 2$ is reducible.

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arXiv2011

Derived Category of Fibrations

Let X ->Y be a Zariski locally trivial fibration of smooth complex projective varieties, with fiber F. We give a structure theorem for the derived category of X provided both F and Z have a full strongly exceptional collection of line bundles.

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arXiv2010

Frobenius splitting and Derived category of toric varieties

The splitting of the Frobenius direct image of line bundles on toric varieties is used to explicitly construct an orthogonal basis of line bundles in the derived category D^b(X) where X is a Fano toric variety with (almost) maximal Picard number.

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arXiv2009

Stability and Unobstructedness of Syzygy Bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,..., d_n}$ on $\PP^N$ defined as the kernel of a general epimorphism $\xymatrix{ϕ:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles $E_{d,n}$ on $\PP^N$ associated to $n$ generic forms $f_1,...,f_n\in K[X_0,X_1,..., X_N]$ of the same degree $d$. Our first goal is to prove that $E_{d,n}$ is stable if $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves, in general, the bound $n\le d(N+1)$ given by G. Hein in \cite{B}, Appendix A. In the last part of the paper, we study moduli spaces of stable rank $n-1$ vector bundles on $\PP^N$ containing syzygy bundles. We prove that if $N+1\le n\le\tbinom{d+2}{2}+N-2$ and $N\ne 3$, then the syzygy bundle $E_{d,n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\tbinom{d+N}{N}-n^2$, if $N \geq 4$, and $n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2$, if N=2.

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arXiv2008

Brill-Noether theory for moduli spaces of sheaves on algebraic varieties

Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for $i=1, ..., s:=min\{r,n\}$. We define the Brill-Noether filtration on $M_{X,H}(r;c_1, ..., c_{s})$ as $W_{H}^{k}(r;c_1,..., c_{s})= \{E \in M_{X,H}(r;c_1, ..., c_{s}) | h^0(X,E) \geq k \}$ and we realize $W_{H}^{k}(r;c_1,..., c_{s})$ as the $k$th determinantal variety of a morphism of vector bundles on $M_{X,H}(r;c_1, ..., c_{s})$, provided $H^i(E)=0$ for $i \geq 2$ and $E \in M_{X,H}(r;c_1, ..., c_{s})$. We also compute the expected dimension of $W_{H}^{k}(r;c_1,..., c_{s})$. Very surprisingly we will see that the Brill-Noether stratification allow us to compare moduli spaces of vector bundles on Hirzebruch surfaces stables with respect to different polarizations. We will also study the Brill-Noether loci of the moduli space of instanton bundles and we will see that they have the expected dimension.

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arXiv2006

Cohomological characterization of vector bundles on multiprojective spaces

We show that Horrock's criterion for the splitting of vector bundles on $\PP^n$ can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool we use the theory of $n$-blocks and Beilinson's type spectral sequences. Cohomological characterizations of vector bundles are also showed.

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arXiv2006

m-blocks collections and Castelnuovo-Mumford regularity in multiprojective spaces

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on $n$-dimensional smooth projective varieties $X$ with an $n$-block collection $\cB $ which generates the bounded derived category $\cD ^b({\cO}_X$-$mod)$. To this end, we use the theory of $n$-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf $F$ on $X$ with respect to the $n$-block collection $\cB $. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on $\PP^n$ and for the $n$-block collection $\cB =(\cO_{\PP^n},\cO_{\PP^n} (1), ..., \cO_{\PP^n}(n))$ on $\PP^n$ Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space $\PP^{n_1}\times ... \times \PP^{n_r}$ with respect to a suitable $n_1+... +n_r$-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang.

math.AG↗
arXiv2006

Geometric collections and Castelnuovo-Mumford Regularity

The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties $X$ with a geometric collection $σ$. We define the notion of regularity of a coherent sheaf $\cF$ on $X$ with respect to $σ$. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $\PP^n$ and for a suitable geometric collection of coherent sheaves on $\PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n \subset \PP^{n+1}$ ($n$ odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo-Mumford regularity of their extension by zero in $\PP^{n+1}$.

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arXiv2002

Ideals of general forms and the ubiquity of the Weak Lefschetz property

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of $R/I$ has been conjectured by Fröberg. In a previous work the authors showed that in many situations the minimal free resolution of $R/I$ must have redundant terms which are not forced by Koszul (first or higher) syzygies among the $F_i$ (and hence could not be predicted from the Hilbert function), but the only examples came when $r=n+1$. Our second main set of results in this paper show that further examples can be obtained when $n+1 \leq r \leq 2n-2$. We also show that if Fröberg's conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Related to the Fröberg conjecture is the notion of Weak Lefschetz property. We continue the description of the ubiquity of this property. We show that any ideal of general forms in $k[x_1,x_2,x_3,x_4]$ has it. Then we show that for certain choices of degrees, any complete intersection has it and any almost complete intersection has it. Finally, we show that most of the time Artinian ``hypersurface sections'' of zeroschemes have it.

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