arXiv subjects

Iustin Coanda

Publications and source records attributed to Iustin Coanda.

15 recordsLinked to original sources

arXiv2024

On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$

The spectrum of a stable rank 2 vector bundle $E$ with $c_1 = 0$ on the projective 3-space is a finite sequence of positive integers $s(0)$, ..., $s(m)$ characterizing the Hilbert function of the graded $H^1$-module of $E$ in negative degrees. Hartshorne [Invent. Math. 66 (1982), 165-190] showed that if $s(i) = 1$ for some $i > 0$ then $s(i+1) = 1$, ..., $s(m) = 1$. We show that if $s(0) = 1$ then $E(1)$ has a global section whose zero scheme is a double structure on a space curve. We deduce, then, the existence of sequences satisfying Hartshorne's condition that cannot be the spectrum of any stable 2-bundle. This provides a negative answer to a question of Hartshorne and Rao [J. Math. Kyoto Univ. 31 (1991), 789-806].

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arXiv2022

Stable rank 3 vector bundles on $\mathbb{P}^3$ with $c_1 = 0$, $c_2 = 3$

We clarify the undecided case $c_2 = 3$ of a theorem of Ein, Hartshorne and Vogelaar [Math. Ann. 259 (1982), 541--569] about the restriction of a stable rank 3 vector bundle with $c_1 = 0$ on the projective 3-space to a general plane. It turns out that there are more exceptions to the stable restriction property than those conjectured by the three authors. One of them is a Schwarzenberger bundle (twisted by $-1$); it has $c_3 = 6$. There are also some exceptions with $c_3 = 2$ (plus, of course, their duals). We also prove, for completeness, the basic properties of the corresponding moduli spaces; they are all nonsingular and connected, of dimension 28.

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arXiv2021

On the vector bundles from Chang and Ran's proof of the unirationality of $\mathcal{M}_g$, $g \leq 13$

We combine the idea of Chang and Ran [Invent. Math. 76 (1984), 41-54] of using monads of vector bundles on the projective 3-space to prove the unirationality of the moduli spaces of curves of low genus with our classification of globally generated vector bundles with small first Chern class $c_1$ on the projective 3-space to get an alternative argument for the unirationality of the moduli spaces of curves of degree at most 13 (based on the general framework of Chang and Ran).

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arXiv2020

Globally generated vector bundles with $c_1 = 5$ on $\mathbb{P}^n$, $n \geq 4$

We complete the classification of globally generated vector bundles with small $c_1$ on projective spaces by treating the case $c_1 = 5$ on $\mathbb{P}^n$, $n \geq 4$ (the case $c_1 \leq 3$ has been considered by Sierra and Ugaglia, while the cases $c_1 = 4$ on any projective space and $c_1 = 5$ on $\mathbb{P}^2$ and $\mathbb{P}^3$ have been studied in two of our previous papers). It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on $\mathbb{P}^5$, and its restriction to $\mathbb{P}^4$. We recall, in an appendix, from our preprint [arXiv:1805.11336], the main results allowing the classification of globally generated vector bundles with $c_1 = 5$ on $\mathbb{P}^3$. Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to $\mathbb{P}^4$ as globally generated vector bundles.

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arXiv2018

Globally generated vector bundles with $c_1 = 5$ on $\mathbb{P}^3$

We provide a classification of globally generated vector bundles with $c_1 = 5$ on the projective 3-space. The classification is complete (except for one case) but not as detailed as the corresponding classification in the case $c_1 = 4$ from our paper [Memoirs A.M.S., Vol. 253, No. 1209 (2018), also arXiv:1305.3464]. We determine, at least, the pairs of integers $(a , b)$ for which there exist globally generated vector bundles on the projective 3-space with Chern classes $c_1 = 5$, $c_2 = a$, $c_3 = b$ (except for the case $(12 , 0)$ and the complementary case $(13 , 5)$ which remain undecided), we describe the Horrocks monads of these vector bundles and we organize them into several families with irreducible bases. We use some of the results from our paper [arXiv:1502.05553] (for which we give, however, a direct self-contained proof in one of the appendices of the present paper) to reduce the problem to the classification of stable rank 3 vector bundles $F$ with $c_1(F) = -1$, $2 \leq c_2(F) \leq 4$, having the property that $F(2)$ is globally generated. We use, then, the spectrum of such a bundle to get the necessary cohomological information. Some of the constructions appearing in the present paper are used (and reproduced, for the reader's convenience) in another paper of ours [arXiv:1711.06060] in which we provide an alternative to Chang and Ran's proof of the unirationality of the moduli spaces of curves of degree at most 13 from [Invent. Math. 76 (1984), 41--54].

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arXiv2016

Globally Generated Vector Bundles on P^n with c_1=4

We classify globally generated vector bundles on the projective n-space with first Chern class = 4. This extends previous results for first Chern class at most 3, namely for 2 of Sierra and Ugaglia [J. Pure Appl. Algebra 213 (2009), 2141-2146] and for 3 of Anghel and Manolache [arXiv:1202.6261] and, independently, of Sierra and Ugaglia [arXiv:1203.0185]. It turns out that the case first Chern class = 4 is much more involved than the previous cases, especially on the projective 3-space. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the projective 4-space (suitably twisted). In the new version Sections 1, 2, 4. 5, 6 and 7 have been rewritten and some arguments and the presentation have been, hopefully, improved.

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arXiv2016

Locally Cohen-Macaulay space curves defined by cubic equations and globally generated vector bundles

We classify globally generated vector bundles with first Chern class $c_1$ at least 4 on the projective 3-space with the property that $E(-c_1+3)$ has a non-zero global section. This (seemingly) technical result allows one to reduce the classification of globally generated vector bundles with $c_1$ at most 7 on the projective 3-space to the classification of stable rank-2 reflexive sheaves with the same properties. The proof is based on a description of the monads of all locally Cohen-Macaulay space curves defined by cubic equations. We extend then this kind of classification to higher dimensional projective spaces. We use this extension to recuperate quickly the classification of globally generated vector bundles with $c_1=4$ on the projective $n$-space for $n$ at least 4, which is part of the main result of our previous paper [arxiv:1305.3464]. We provide, in the appendices to the paper, graded free resolutions for the homogeneous ideals and for the graded structural algebras of all non-reduced locally Cohen-Macaulay space curves of degree at most 4.

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arXiv2016

Four generated 4-instantons

We show that there exist mathematical 4-instanton bundles F on the projective 3-space such that F(2) is globally generated (by four global sections). This is equivalent to the existence of elliptic space curves of degree 8 defined by quartic equations. There is a (possibly incomplete) intersection theoretic argument for the existence of such curves in D'Almeida [Bull. Soc. Math. France 128 (2000), 577-584] and another argument, using results of Mori [Nagoya Math. J. 96 (1984), 127-132], in Chiodera and Ellia [Rend. Istit. Univ. Trieste 44 (2012), 413-422]. Our argument is quite different. We prove directly the former fact, using the method of Hartshorne and Hirschowitz [Ann. Scient. Ec. Norm. Sup. (4) 15 (1982), 365-390] and the geometry of five lines in the projective 3-space.

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arXiv2012

A refined stable restriction theorem for vector bundles on quadric threefolds

Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a closed subset of codimension at least 2 of the dual projective 4-space, and we explicitly describe the bundles E which do not enjoy this property. This refines a restriction theorem of Ein and Sols [Nagoya Math. J. 96, 11-22 (1984)] in the same way the main result of Coanda [J. reine angew. Math. 428, 97-110 (1992)] refines the restriction theorem of Barth [Math. Ann. 226, 125-150 (1977)].

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arXiv2010

A simple proof of Tyurin's babylonian tower theorem

Using the method of Coandă and Trautmann (2006), we give a simple proof of the following theorem due to Tyurin (1976) in the smooth case: if a vector bundle $E$ on a $c$-codimensional locally Cohen-Macaulay closed subscheme $X$ of the projective space $P^n$ extends to a vector bundle $F$ on a similar closed subscheme $Y$ of $P^N$, for every $N > n$, then $E$ is the restriction to $X$ of a direct sum of line bundles on $P^n$. Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.

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arXiv2010

The Horrocks correspondence for coherent sheaves on projective spaces

We establish an equivalence between the stable category of coherent sheaves (satisfying a mild restriction) on a projective space and the homotopy category of a certain class of minimal complexes of free modules over the exterior algebra Koszul dual to the homogeneous coordinate algebra of the projective space. We also relate these complexes to the Tate resolutions of the respective sheaves. In this way, we extend from vector bundles to coherent sheaves the results of Coandă and Trautmann [Trans. AMS 385 (2005)], which interpret in terms of the BGG correspondence the results of Trautmann [Math. Ann. 237 (1978)] about the correspondence of Horrocks [Proc. London. Math. Soc. 14 (1964)], [Asterisque 71-72 (1980)]. We also give direct proofs of the BGG correspondences for graded modules and for coherent sheaves and of the theorem of Eisenbud, Floystad and Schreyer [Trans. AMS 355 (2003)] describing the linear part of the Tate resolution associated to a coherent sheaf. Moreover, we provide an explicit description of the quotient of the Tate resolution by its linear strand corresponding to the module of global sections of the various twists of the sheaf.

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arXiv2009

On the stability of syzygy bundles

We are concerned with the problem of the stability of the syzygy bundles associated to base point free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from Mark Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if his rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner (1984). We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n=2, by L. Costa, P. Macias Marques and R.M. Miro-Roig (2009). The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis (2009).

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arXiv2009

Infinitely stably extendable vector bundles on projective spaces

According to Horrocks (1966), a vector bundle E on the projective n-space extends stably to the projective N-space, N>n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a direct sum of line bundles. We show that E extends stably to the projective N-space for every N>n if and only if E is the cohomology of a free monad (with three terms). The proof uses the method of Coanda and Trautmann (2006). Combining this result with a theorem of Mohan Kumar, Peterson and Rao (2003), we get a new effective version of the Babylonian tower theorem for vector bundles on projective spaces.

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arXiv2004

The splitting criterion of Kempf and the Babylonian tower theorem

We show that the idea used by Kempf (1990) in order to obtain a splitting criterion for vector bundles on projective spaces leads to an elementary proof of the Babylonian tower theorem for this class of bundles, a result due to Barth--Van de Ven (1974) in the rank 2 case and to Sato (1977) and Tyurin (1976) in the case of arbitrary rank. As a byproduct we obtain a slight improvement of the numerical criterion of Flenner (1985) in the particular case under consideration.

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arXiv2002

On the Bernstein-Gel'fand-Gel'fand correspondence and a result of Eisenbud, Fløystad, and Schreyer

We show that a combination between a remark from the well known note of I.N. Bernstein, I.M. Gel'fand and S.I. Gel'fand and the idea, systematically investigated in a recent work of D. Eisenbud, G. Fløystad and F.-O. Schreyer, of taking Tate resolution over exterior algebras leads to quick proofs of the main results of these two papers. This combination is expressed by a lemma which we prove directly using the cohomology of invertible sheaves on a projective space.

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