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Cristian Anghel

Publications and source records attributed to Cristian Anghel.

16 recordsLinked to original sources
arXiv2026

Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem

In their 2003 article in the Journal of the American Mathematical Society, Eisenbud, Schreyer and Weyman asked whether every embedded projective variety carries an Ulrich sheaf. In 2017 Beauville proposed a numerical route toward a surface with no Ulrich bundles: in Picard rank one, existence forces $H^2\ge K_S^2-8\chi(\mathcal O_S)$, suggesting a search near the Bogomolov--Miyaoka--Yau boundary. We show that the Picard-rank-one hypothesis is not needed for the obstruction: a rank-independent Bogomolov--Hodge argument gives the same inequality on every smooth polarized surface. Using additional Neron--Severi directions on Hirzebruch--Kummer resolutions, the exponent-$3$ Hesse surface admits a very ample class $H=4A-E$ with $H^2=7\cdot3^9<16\cdot3^9=K_Y^2-8\chi(\mathcal O_Y)$, hence a smooth counterexample to the Eisenbud--Schreyer--Weyman problem in its standard formulation. Consequently its Chow form has no ESW-type linear determinantal representation arising from an Ulrich sheaf on the embedded surface. Moreover, for every $n\ge3$ the Hesse pair $(Y_n,4A_n-E_n)$ is a counterexample, the surfaces are pairwise non-isomorphic, and $H_n^2/\sigma(Y_n)=7/(3n^2-11)\to0$, while $K_{Y_n}^2/\chi(\mathcal O_{Y_n})\to60/7\approx8.5714$. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.

math.AG↗
arXiv2026

A no-go theorem for special Ulrich bundles, with a complement on primary Burniat surfaces

Let $X$ be a smooth projective surface with $p_g=0$, and let $H$ be an ample divisor with $h^0(\mathcal{O}_X(H))\neq0$, $\chi(\mathcal{O}_X(H))\ge q$, and $h^1(\mathcal{O}_X(H))\neq0$. We prove that no rank two bundle $\mathcal{E}$ with $c_1(\mathcal{E})=3H+K_X$, with the Ulrich value of $c_2(\mathcal{E})$, and satisfying $h^0(\mathcal{E}(-H))=0$, can arise from an extension $0 \to \mathcal{O}_X(H+K_X) \to \mathcal{E} \to \mathcal{O}_X(2H)\otimes\mathcal{I}_Z \to 0$. Thus, for this natural Cayley-Bacharach construction, non-speciality of the polarization is necessary rather than merely convenient. We then study primary Burniat surfaces. We show that every ample and base point free divisor is non-special; consequently, every polarization carries a stable special Ulrich bundle of rank two, and the surface is strictly Ulrich wild with respect to every polarization. We also locate the special ample classes on three numerical rays through $K_X$, compute explicit families on these rays, and analyze a twisted-kernel variant of the construction. The degree bound underlying the non-speciality result overlaps with recent work of Y. Cho, while the global consequences and the no-go theorem are independent.

math.AG↗
arXiv2026

AKSZ Descent on Manifolds with Ordinary Corners

Under an explicit formal mapping-space hypothesis, we develop a facewise formulation of the classical AKSZ construction on compact oriented manifolds with ordinary corners. The codimension-$r$ data---a mapping space carrying a closed two-form of degree $r-1$, an action of degree $r$, and a cohomological vector field---and the modified Batalin--Vilkovisky/Batalin--Fradkin--Vilkovisky Hamiltonian identity relating consecutive strata are those of the maximally extended BV--BFV theory of Cattaneo--Mnev--Reshetikhin. What is added here is the organization over the entire face poset: the Hamiltonian defect on a face is the sum of the pullbacks of the primitives on its codimension-one faces, weighted by the orientation incidence numbers, so that the boundary term of the single-stratum identity is resolved into its connected pieces with signs. Organizing these defects by the face incidence complex yields a total-complex theorem: factorially normalized facewise transgression is a cochain map, so closed target forms transgress to cocycles, and the twice-iterated defect vanishes because the signed face differential squares to zero. We verify all four codimension-two cancellations explicitly for four-dimensional BF theory on $M=\Gamma\times[0,1]^2$. We also establish a reduction criterion for singular corner data. If a raw codimension-two descendant is presymplectic and its reduced Dirac structure is the graph of a Poisson bivector, the shifted cotangent construction gives a canonical strict degree-two corner theory. The passage from a reduced Poisson bivector to a strict corner theory is already recorded in \cite{CFT2026}; what is isolated here is the hypothesis under which it applies, and its relation to the face-incidence structure. The construction provides a rigorous ordinary-corner benchmark for extensions of AKSZ descent to Joyce generalized corners.

math.SG↗
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).

math.AG↗
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.

math.AG↗
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].

math.AG↗
arXiv2017

Geometry of the Sasakura bundle

The Sasakura bundle is a relatively recent appearance in the world of remarkable vector bundles on projective spaces. In fact, it is connected with some surfaces in $\mathbb P^4$ which missed in early classification papers. The aim of this note is to present various, scattered in the literature, aspects concerning the geometry of this bundle. The last part will be devoted to the place of this bundle in the classification of globally generated locally free sheaves with $c_1 \leq 4$ on $\mathbb P^n$ in a joint paper with I. Coanda and N. Manolache.

math.AG↗
arXiv2016

A stable version of Terao conjecture

The aim of this note is to introduce a stable version of Terao conjecture, using the notion of infinitely stably extendability of vector bundles on $\mathbb P^n$, considered and characterized by I. Coanda in arXiv:0907.4040.

math.AG↗
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.

math.AG↗
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.

math.AG↗
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.

math.AG↗
arXiv2015

Quantum Sheaf Cohomology on surfaces of general type I: construction of stable omalous bundles

Quantum sheaf cohomology is a deformation of the cohomology ring of a sheaf. In recent years, this subject had an impetuous development in connection with the $(0; 2)$ non-linear sigma model from super-strings theory. The basic piece in this area is a so-called omalous bundle on the variety we start with. After a short overview of the subject, we construct stable omalous bundles on some classes of surfaces of general type.

math.AG↗
arXiv2012

Globally Generated Vector Bundles on P^n with c_1=3

One classifies the globally generated vector bundles on P^n (n \not = 3) with the first Chern class c_1 = 3. The case n = 3 is treated in arXiv:1202.5988 [math.AG]. The case c_1 = 2 was treated by J.C. Sierra and L. Ugaglia (see References), the case c_1 = 3, rank = 2 is settled by S. Huh (see References), the case rank = 2, c_1 \le 5 is studied by L. Chiodera and Ph. Ellia (see References).

math.AG↗
arXiv2009

Development and Optimization of a Multimedia Product

This article presents a new concept of a multimedia interactive product. It is a multiuser versatile platform that can be used for different purposes. The first implementation of the platform is a multiplayer game called Texas Hold 'em, which is a very popular community card game. The paper shows the product's multimedia structure where Hardware and Software work together in creating a realistic feeling for the users.

cs.MM↗
arXiv2001

Complete subvarieties in moduli spaces of rank 2 stables sheaves on smooth projective curves and surfaces

The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary $c_1$ and sufficiently large $c_2$ on algebraic surfaces. Then we study the restriction of these sheaves to curves of high degree embedded in the surface. In the final section we gives a relation with the spin strata defined by Pidstrigach and Tyurin.

math.AG↗