Marcello Bernardara

1. Even nodal surfaces of K3 type (with E. Fatighenti, G. Kapustka, M. Kapustka, G. Mongardi, L. Manivel and F. Tanturri), preprint: ArXiv:2402.08528.
We study Fano fourfolds of K3 type with a conic bundle structure. We construct direct geometrical links between these fourfolds and hyperKähler varieties. As a result we describe families of nodal surfaces that can be seen as generalisations of Kummer quartic surfaces. Each of these families actually arises through two families of Fano fourfolds, whose conic bundle structures are related by hyperbolic reduction.


2. Fano fourfolds of K3 type (with E. Fatighenti, L. Manivel and F. Tanturri), preprint: ArXiv:2111.13030.
We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of flag manifolds. We study the geometry of these Fano fourfolds in some detail, and we find the origin of their K3 structure by relating most of them either to cubic fourfolds, Gushel-Mukai fourfolds, or actual K3 surfaces. Their main invariants and some information on their rationality and on possible semiorthogonal decompositions for their derived categories are provided.


3. Nested varieties of K3 type (with E. Fatighenti and L. Manivel), J. Éc. Polytech., Math. 8, 733-778 (2021).
Using geometrical correspondences induced by projections and two-steps flag varieties, and a generalization of Orlov's projective bundle theorem, we relate the Hodge structures and derived categories of subvarieties of different Grassmannians. We construct isomorphisms between Calabi-Yau subHodge structures of hyperplane sections of Gr(3,n) and those of other varieties arising from symplectic Grassmannian and/or congruences of lines or planes. Similar results hold conjecturally for Calabi-Yau subcategories: we describe in details the Hodge structures and give partial categorical results relating the K3 Fano hyperplane sections of Gr(3,10) to other Fano varieties such as the Peskine variety. Moreover, we show how these correspondences allow to construct crepant categorical resolutions of the Coble cubics.


4. A categorical invariant for geometrically rational surfaces with a conic bundle structure (with S. Durighetto), Progr. Math., 342, Birkhäuser/Springer, Cham, 2021, 113--128. .
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces.


5. Categorical dimension of birational automorphisms and filtrations of Cremona groups, Journal of the Math. Soc. of Japan 73, No. 3, 861-883 (2021).
Using a filtration on the Grothendieck ring of triangulated categories, we define the categorical dimension of a birational map between smooth projective varieties. We show that birational automorphisms of bounded categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. In the case of threefolds, we can moreover recover the genus of a birational automorphism, and the filtration defined by Frumkin.


6. Cycles, derived categories, and rationality (with A. Auel), in Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics 95, 199-266 (2017).
Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these perspectives intertwine and reflect each other. In particular, in the case of algebraic surfaces, we explain the relationship between Bloch's conjecture, Chow-theoretic decompositions of the diagonal, categorical representability, and the existence of phantom subcategories of the derived category.


7. Bridgeland Stability Conditions on Fano Threefolds (with E. Macrì, B. Schmidt, and X. Zhao), Épijournal de Géométrie Algébrique Vol 1 (2017).
We show the existence of Bridgeland stability conditions on all Fano threefolds, by proving a modified version of a conjecture by Bayer, Toda, and the second author. The key technical ingredient is a strong Bogomolov inequality, proved recently by Chunyi Li. Additionally, we prove the original conjecture for some toric threefolds by using the toric Frobenius morphism.


8. Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields (with A. Auel), Proc. London Math. Soc. (3) 117 (2018) 1-64.
We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern classes associated to these vector bundles.


9. Homological projective duality for determinantal varieties (with M.Bolognesi and D.Faenzi), Adv. Math. 296, 181-209 (2016).
In this paper we prove Homological Projective Duality for crepant categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a \(\scriptsize n \times m\) matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety.


10. Some remarks concerning Voevodsky's nilpotence conjecture (with M.Marcolli and G.Tabuada), J. Reine Angew. Math. 738 (2018), 299-312.
In this note we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture is the case of quadric fibrations, intersection of quadrics, linear sections of Grassmannians, and Moishezon manifolds.


11. Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives (with G.Tabuada), Izvestiya: Mathematics 80, no. 3 (2016) , preprint: ArXiv:1310.6020.
Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we compute the rational Chow groups of a complete intersection of either two quadrics or three odd-dimensional quadrics. We show moreover that the unique non-trivial algebraic Jacobians are the middle ones. As a first application, we describe the rational Chow motives of these complete intersections. As a second application, we prove that smooth fibrations in such complete intersections over small dimensional bases S verify Murre's conjecture (\(\scriptsize \mathrm{dim}(S) \leq 1\)), Grothendieck's standard conjectures \(\scriptsize \mathrm{dim}(S) \leq 2\)), and Hodge's conjecture (\(\scriptsize \mathrm{dim}(S) \leq 3\)).


12. From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives (with G.Tabuada), Moscow Math. Journal 16 (2), pp. 205-235 (2016).
Let \(\scriptsize X\) and \(\scriptsize Y\) be complex smooth projective varieties, and \(\scriptsize \mathbf{D}^b(X)\) and \(\scriptsize \mathbf{D}^b(Y)\) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category \(\scriptsize \mathbf{T}\) which is admissible both in \(\scriptsize \mathbf{D}^b(X)\) as in \(\scriptsize \mathbf{D}^b(Y)\). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism \(\scriptsize \tau\) of abelian varieties (up to isogeny) from the product of the intermediate algebraic Jacobians of \(\scriptsize X\) to the product of the intermediate algebraic Jacobians of \(\scriptsize Y\). When the orthogonal complement of \(\scriptsize \mathbf{T}\) in \(\scriptsize \mathbf{D}^b(X)\) has a trivial Jacobian (e.g. when it is generated by exceptional objects), the morphism \(\scriptsize \tau\) is split injective. When this also holds for the orthogonal complement of \(\scriptsize \mathbf{T}\) in \(\scriptsize \mathbf{D}^b(Y)\), \(\scriptsize \tau\) becomes an isomorphism. Furthermore, in the case where \(\scriptsize X\) and \(\scriptsize Y\) have a single intermediate algebraic Jacobian carrying a principal polarization, we prove that the morphism t preserves this extra structure. As an application, we obtain categorical Torelli theorems, an incompatibility between two conjectures of Kuznetsov (one concerning Fourier-Mukai functors and another one concerning Fano threefolds) a new proof of a classical theorem of Clemens-Griffiths on blow-ups of threefolds,and several new results on quadric fibrations and intersection of quadrics.


13. Relations between the Chow motive and the noncommutative motive of a smooth projective variety (with G.Tabuada), Journal of Pure and Applied Algebra 219 (2015), pp. 5068-5077.
In this note we relate the notions of Lefschetz type, decomposability, and isomorphism, on Chow motives with the notions of unit type, decomposability, and isomorphism, on noncommutative motives. Some examples, counter-examples, and applications are also described.


14. Cubic fourfolds containing a plane and a quintic del Pezzo surface (with A. Auel, M. Bolognesi, and A. Várilly-Alvarado), Algebraic Geometry 2 (2014) 181-193.
We isolate a class of smooth rational cubic fourfolds \(\scriptsize X\) containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface \(\scriptsize S\) of degree 2 associated to \(\scriptsize X\). Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors \(\scriptsize \mathcal{C}_8\) and \(\scriptsize \mathcal{C}_{14}\) has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve, we prove that the general member is both pfaffian and has nontrivial Brauer class. Such cubic fourfolds also provide twisted derived equivalences between K3 surfaces of degree 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.


15. Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems (with A. Auel, and M. Bolognesi), Journal de Math. Pures et Appliquées 102, 249-291 (2014).
Let \(\scriptsize X \to Y\) be a fibration whose fibers are complete intersections of \(\scriptsize r\) quadrics. We develop some categorical and algebraic tools, such as relative homological projective duality and the Morita-invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting, to study semiorthogonal decompositions of the bounded derived category of X. Together with input from the theory of quadratic forms, we apply these tools in the case where \(\scriptsize r=2\) and \(\scriptsize X \to Y\) has relative dimension 1, 2, or 3, in which case the fibers are elliptic curves, Del Pezzo surfaces of degree 4, or Fano three-folds, respectively. In the latter two cases, if \(\scriptsize Y\) is the projective line over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of \(\scriptsize X\).


16. Categorical representability and intermediate Jacobians of Fano threefolds (with M. Bolognesi), in Derived Categories in Algebraic Geometry, EMS Series of Congress Reports, 1-25 (2012).
We define, basing upon semiorthogonal decompositions of \(\scriptsize \mathbf{D}^b(X)\), categorical representability of a projective variety \(\scriptsize X\) and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of \(\scriptsize X\) by providing examples and stating open questions.


17. Derived categories and rationality of conic bundles (with M. Bolognesi), Comp.Math 149 (11), 1789-1817 (2013).
We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.


18. The Euclid-Fourier-Mukai algorithm for elliptic surfaces (with G. Hein) Asian J. of Math 18 (2), 345-364 (2014).
We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of \(\scriptsize P\)-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the Picard groups of the moduli spaces.


19. A categorical invariant for cubic threefolds (with E. Macrì, S. Mehrotra, and P. Stellari), Adv. Math. 229 (2), 770-803 (2012).
We prove a categorical version of the Torelli Theorem for cubic threefolds. More precisely, we show that the non trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.


20. Stable pairs on elliptic K3 surfaces, C. R. Acad. Sci. Paris, Ser. I 348 (2010).
We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.


21. Fourier-Mukai transforms of curves and principal polarizations, C. R. Acad. Sci. Paris, Ser. I 345 (2007).
Given a Fourier-Mukai transform \(\scriptsize \Phi\) between the bounded derived categories of two smooth projective curves, we verifiy that the induced map between the Jacobian varieties preserves the principal polarization if and only if \(\scriptsize \Phi\) is an equivalence.


22. A semiorthogonal decomposition for Brauer-Severi schemes, Math. Nachr. 282, No. 10, 1406-1413 (2009).
A semiorthogonal decomposition for the bounded derived category (the category of perfect complexes in a non smooth case) of coherent sheaves on a Brauer-Severi scheme is given. It relies on bounded derived categories (categories of perfect complexes in a non smooth case) of suitably twisted coherent sheaves on the base.


23. Calabi-Yau complete intersections with infinitely many lines, Rend. Semin. Mat. Univ. Politec. Torino, 66 (2008), no 2.
We give two new examples of families of Calabi-Yau complete intersection threefolds whose generic element contains infinitely many lines. We get some results about the normal bundles of these lines and the Hilbert scheme of lines on the threefolds. In particular, the surface swept out by such a family is not a cone.

Theses

• My Habilitation Thesis, Semiorthogonal decompositions and noncommutative motives in algebraic geometry, discussed on 21st October 2016.
• My PhD Thesis, Catégories dérivées, espaces de modules, discussed on 23rd June 2008, under the direction of Arnaud Beauville.

Posters and Beamer presentations

• My Poster at the Cremona Conference in Basel, September 2016.
• A talk about cubic surfaces, 27 lines, and the icosahedron in Courant Institute, New York City, November 2012.
• My Poster at the Nachwuchskonferenz in Mainz, November 2010.