An asymptotic bound for Castelnuovo-Mumford regularity of certain Ext modules over graded complete intersection rings

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

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Heterotic line bundle models on generalized complete intersection Calabi Yau manifolds

Journal of High Energy Physics ◽

10.1007/jhep05(2021)105 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Magdalena Larfors ◽

Davide Passaro ◽

Robin Schneider

Keyword(s):

Line Bundle ◽

Complete Intersection ◽

Particle Physics ◽

Model Building ◽

Wilson Line ◽

Symmetry Groups ◽

The Standard Model ◽

Order Of Magnitude ◽

Systematic Program ◽

Calabi Yau Manifolds

Abstract The systematic program of heterotic line bundle model building has resulted in a wealth of standard-like models (SLM) for particle physics. In this paper, we continue this work in the setting of generalised Complete Intersection Calabi Yau (gCICY) manifolds. Using the gCICYs constructed in ref. [1], we identify two geometries that, when combined with line bundle sums, are directly suitable for heterotic GUT models. We then show that these gCICYs admit freely acting ℤ2 symmetry groups, and are thus amenable to Wilson line breaking of the GUT gauge group to that of the standard model. We proceed to a systematic scan over line bundle sums over these geometries, that result in 99 and 33 SLMs, respectively. For the first class of models, our results may be compared to line bundle models on homotopically equivalent Complete Intersection Calabi Yau manifolds. This shows that the number of realistic configurations is of the same order of magnitude.

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Ulrich Bundles on Intersections of Two 4-Dimensional Quadrics

International Mathematics Research Notices ◽

10.1093/imrn/rnz320 ◽

2019 ◽

Author(s):

Yonghwa Cho ◽

Yeongrak Kim ◽

Kyoung-Seog Lee

Keyword(s):

Complete Intersection ◽

Moduli Space ◽

Main Ingredient ◽

Categorical Methods

Abstract In this paper, we investigate the moduli space of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$. The main ingredient is the semiorthogonal decomposition by Bondal–Orlov, combined with the categorical methods pioneered by Kuznetsov and Lahoz–Macrì–Stellari. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in $\mathbb P^5$ carries an Ulrich bundle of rank $r$ for every $r \ge 2$. Moreover, we provide a description of the moduli space of stable Ulrich bundles.

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ON THE STABILITY OF SYZYGY BUNDLES

International Journal of Mathematics ◽

10.1142/s0129167x1100688x ◽

2011 ◽

Vol 22 (04) ◽

pp. 515-534 ◽

Cited By ~ 1

Author(s):

IUSTIN COANDĂ

Keyword(s):

Projective Space ◽

Complete Intersection ◽

Vector Bundles ◽

Elementary Proof ◽

Base Point ◽

Vector Spaces ◽

Similar Argument ◽

Vanishing Theorem ◽

The Stability ◽

Koszul Cohomology

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 M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its 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 [Comment. Math. Helv.59 (1984) 635–650]. 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 [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

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