Michele Bolognesi | Université de Montpellier (original) (raw)
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Papers by Michele Bolognesi
Forum of mathematics. Sigma, 2024
arXiv (Cornell University), Aug 21, 2023
arXiv (Cornell University), Mar 5, 2009
In this paper we explore the intersection of the Hassett divisor C8, parametrizing smooth cubic f... more In this paper we explore the intersection of the Hassett divisor C8, parametrizing smooth cubic fourfolds X containing a plane P with other divisors Ci. Notably we study the irreducible components of the intersections with C12 and C20. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of P with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off P , or by finding examples of reducible one-apparent-double-point surfaces inside X. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.
Transactions of the American Mathematical Society, Jul 1, 2012
arXiv (Cornell University), Oct 13, 2017
arXiv (Cornell University), Feb 18, 2007
European journal of mathematics, May 9, 2022
arXiv (Cornell University), Jul 12, 2023
Let X^{1,n}_r be the blow-up of P1 × Pn in r general points. We describe the Mori cone of X^{1,n}... more Let X^{1,n}_r be the blow-up of P1 × Pn in r general points. We describe the Mori cone of X^{1,n}_r for r ≤ n + 2 and for r = n + 3 when n ≤ 4. Furthermore, we prove that X^{1,n}_ n+1 is log Fano and give an explicit presentation for its Cox ring.
Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with ... more Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in P^8as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and
Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU_C(2,L), the moduli space of rank two stable vector bundles
on C with fixed determinant of odd degree L, as a subvariety of G(2, 8). In fact, each point p ∈ C defines a natural embedding of SU_C(2,O(p)) in G(2, 8). We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of SU_C(2,O(p)), and thus deserves to be coined the Coble
quadric of the pointed curve (C, p).
Cornell University - arXiv, Sep 26, 2022
We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatig... more We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatighenti, Manivel and Tanturri in [6], have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. It follows that the Chow ring of these Fano varieties behaves like that of K3 surfaces. As a side result, we obtain some criteria for the Franchetta property of blown-up projective varieties.
ABSTRACT In this paper we study the topology of the stack mathcalTg\mathcal{T}_gmathcalTg of smooth trigonal curv... more ABSTRACT In this paper we study the topology of the stack mathcalTg\mathcal{T}_gmathcalTg of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes mathcalTg\mathcal{T}_gmathcalTg as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of mathcalTg\mathcal{T}_gmathcalTg, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group mathcalMapg\mathcal{M}ap_gmathcalMapg of Riemann surfaces of genus g.
Journal of Pure and Applied Algebra, Aug 1, 2020
Gushel-Mukai sixfolds are an important class of so-called FanoK3 varieties. In this paper we show... more Gushel-Mukai sixfolds are an important class of so-called FanoK3 varieties. In this paper we show that they admit a multiplicative ChowKünneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double EPW sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of Gushel-Mukai sixfolds.
We define universal Gushel-Mukai fourfolds over certain Noether-Lefschetz loci in the moduli stac... more We define universal Gushel-Mukai fourfolds over certain Noether-Lefschetz loci in the moduli stack of Gushel-Mukai fourfolds ℳ^4_GM. Using the relation between these fourfolds and K3 surfaces, we relate moduli of K3 surfaces to universal Gushel-Mukai varieties, and their birational geometry. This allows us to prove the unirationality or rationality of some of these universal families.
We give an algebro-geometric construction of the Hitchin connection, valid also in positive chara... more We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott K\"ahler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson-Schechtman and Bloch-Esnault.
Forum of mathematics. Sigma, 2024
arXiv (Cornell University), Aug 21, 2023
arXiv (Cornell University), Mar 5, 2009
In this paper we explore the intersection of the Hassett divisor C8, parametrizing smooth cubic f... more In this paper we explore the intersection of the Hassett divisor C8, parametrizing smooth cubic fourfolds X containing a plane P with other divisors Ci. Notably we study the irreducible components of the intersections with C12 and C20. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of P with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off P , or by finding examples of reducible one-apparent-double-point surfaces inside X. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.
Transactions of the American Mathematical Society, Jul 1, 2012
arXiv (Cornell University), Oct 13, 2017
arXiv (Cornell University), Feb 18, 2007
European journal of mathematics, May 9, 2022
arXiv (Cornell University), Jul 12, 2023
Let X^{1,n}_r be the blow-up of P1 × Pn in r general points. We describe the Mori cone of X^{1,n}... more Let X^{1,n}_r be the blow-up of P1 × Pn in r general points. We describe the Mori cone of X^{1,n}_r for r ≤ n + 2 and for r = n + 3 when n ≤ 4. Furthermore, we prove that X^{1,n}_ n+1 is log Fano and give an explicit presentation for its Cox ring.
Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with ... more Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in P^8as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and
Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU_C(2,L), the moduli space of rank two stable vector bundles
on C with fixed determinant of odd degree L, as a subvariety of G(2, 8). In fact, each point p ∈ C defines a natural embedding of SU_C(2,O(p)) in G(2, 8). We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of SU_C(2,O(p)), and thus deserves to be coined the Coble
quadric of the pointed curve (C, p).
Cornell University - arXiv, Sep 26, 2022
We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatig... more We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatighenti, Manivel and Tanturri in [6], have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. It follows that the Chow ring of these Fano varieties behaves like that of K3 surfaces. As a side result, we obtain some criteria for the Franchetta property of blown-up projective varieties.
ABSTRACT In this paper we study the topology of the stack mathcalTg\mathcal{T}_gmathcalTg of smooth trigonal curv... more ABSTRACT In this paper we study the topology of the stack mathcalTg\mathcal{T}_gmathcalTg of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes mathcalTg\mathcal{T}_gmathcalTg as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of mathcalTg\mathcal{T}_gmathcalTg, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group mathcalMapg\mathcal{M}ap_gmathcalMapg of Riemann surfaces of genus g.
Journal of Pure and Applied Algebra, Aug 1, 2020
Gushel-Mukai sixfolds are an important class of so-called FanoK3 varieties. In this paper we show... more Gushel-Mukai sixfolds are an important class of so-called FanoK3 varieties. In this paper we show that they admit a multiplicative ChowKünneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double EPW sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of Gushel-Mukai sixfolds.
We define universal Gushel-Mukai fourfolds over certain Noether-Lefschetz loci in the moduli stac... more We define universal Gushel-Mukai fourfolds over certain Noether-Lefschetz loci in the moduli stack of Gushel-Mukai fourfolds ℳ^4_GM. Using the relation between these fourfolds and K3 surfaces, we relate moduli of K3 surfaces to universal Gushel-Mukai varieties, and their birational geometry. This allows us to prove the unirationality or rationality of some of these universal families.
We give an algebro-geometric construction of the Hitchin connection, valid also in positive chara... more We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott K\"ahler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson-Schechtman and Bloch-Esnault.
Starting from known 4-dimensional families of K3 surfaces, we construct two families of cubic fou... more Starting from known 4-dimensional families of K3 surfaces, we construct two families of cubic fourfolds whose motive is of Abelian type. Cubics from the first family are smooth, and their Chow motive is finite dimensional and Abelian. Those from the second family are singular, and their motives are Schur-finite and Abelian in Voevodsky's triangulated category of motives.
The aim of this short note is to de ne the universal cubic fourfold over certain loci of the modu... more The aim of this short note is to de ne the universal cubic fourfold
over certain loci of the moduli space. Then, by using some recent results of Farkas-Verra and more classical ones by Mukai, we prove that the universal cubic fourfolds over the Hassett divisors C14, C26 and C42 are unirational. As for the rationality of cubic fourfolds, this relies heavily on the existence of associated K3 surfaces, and the birational geometry of their universal families.
Finally, we observe that for explicit in nitely many values of d, the universal cubic fourfold over Cd can not be unirational.