Splitting, construction and stability of vector bundles and their applications
Dates：June 20-June 21, 2018
Place：Kyushu University, Ito Campus, West-1 C515
Speakers：
Takeshi Abe（Kumamoto University）
Hirotachi Abo （Idaho University）
Daniele Faenzi (University of Bourgogne)
Hajime Kaji （Waseda University）
Simone Marchesi (IMECC-UNICAMP)
Chikashi Miyazaki（Kumamoto University）
Joan Pons-Llopis (University of L'aquila)
Eiichi Sato（Kyushu University）
Jean Valles (University of Pau and Pays de l'Adour)
Kenta Watanabe (Nihon University)
Program
20 June (Wednesday)
9:30 -- 10:30 Daniele Faenzi (University of Bourgogne)
Tite:Vector bundles on the space of forms
10:45 -- 11:45 Kenta Watanabe (Nihon University)
Title: On the classification of rank two ACM bundles on quartic hypersurfaces in $\mathbb{P}^3$
13:30 -- 14:30 Joan Pons-Llopis (University of l'Aquila)
Title: The Minimal Resolution Conjecture (MRC) for points on the cubic scroll
15:00 -- 16:00 Takeshi Abe (Kumamoto University)
Title: Semistable sheaves with symmetric $c_{1}$ on a quadric surface
16:15 -- 17:15 Hajime Kaji (Waseda University)
Title: Degree formula for Grassmann bundles and its applications
21 June (Thursday)
9:30 -- 10:30 Jean Valles (University of Pau and Pays de l'Adour)
Title: Unexpected divisors of jumping lines
10:45 -- 11:45 Simone Marchesi (IMECC-UNICAMP)
Title: Uniform Steiner bundles
13:30 -- 14:30 Hirotachi Abo (University of Idaho)
Title: On the discriminant locus of a vector bundle
15:00 -- 16:00 Eiichi Sato (Kyushu University)
Title: On Fano bundles of higher rank on the projective spaces
16:15 -- 17:15 Chikashi Miyazaki （Kumamoto University)
Title: Cohomological Criteria for Vector Bundles on Multiprojective Space
Abstracts:
Takeshi Abe (Kumamoto University）
Title: Semistable sheaves with symmetric $c_{1}$ on a quadric surface
Abstract: For moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface, we pursue analogy to some results known for moduli spaces of sheaves on a projective plane. We dene an invariant height, introduced by Drezet in the projective plane case, for moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface and describe the structure of moduli spaces of height zero. If time permits, we also talk about effective cones of moduli spaces, and strange duality for height-zero moduli spaces.
The content of the talk has already been published in Nagoya Math J. Vol 227 (2017).
Hirotachi Abo（University of Idaho）
Title: On the discriminant locus of a vector bundle
Abstract:The main purpose of the talks is to discuss the variety formed by global sections of a vector bundle on projective space whose zero scheme is singular. We call such a variety the discriminant locus of a global section of the vector bundle (or just the discriminant of the vector bundle).
The discriminant locus of a vector bundle can be very naturally considered as a generalization of the classical discriminant of a polynomial in one variable, i.e., a polynomial in the coefficients which vanishes at the polynomial whenever it has a multiple root. It is very natural to ask ”What is the dimension of the discriminant of a vector bundle? What about the degree?” The discriminant locus of a line bundle is also naturally identified with the dual variety of the Veronese variety, and hence its dimension and degree are already known. The focus of this talk is therefore on the discriminant locus for a vector bundle of higher rank. In this talk, I plan to show that the discriminant locus of a so-called the Schwarzenberger type bundle (STB) is an irreducible hypersurface. I also plan to discuss how the geometry of a non-singular curveassociated with STB helps us find the degree of such a hypersurface.
Daniele Faenzi（University of Bourgogne）
Title: Vector bundles on the space of forms
Abstract: On the projective space $\mathbb{P}^n$, say for $n > 3$, vector bundles of rank $r < n$ are difficult to find. For $n >5$ and $r < n-1$, no such bundle is known. A couple of well-known results constructions provide bundles of rank $r = n-1$ (Tango bundles, instanton bundles) and allow to define new bundles starting from these; but that is all we know up to now.
In this talk, I will present a construction providing new bundles of rank $r = n-1$, homogeneous for the action of $\mbox{SL}_2$ operating by the action on binary forms. The construction generalizes to give $\mbox{SL}_m$-homogennous bundles on the space of symmetric forms of degree $d$ in $m$ variables.
Hajime Kaji（Waseda University）
Title: Degree formula for Grassmann bundles and its applications
Abstract: The Grassmann bundle $\mathbb G_{X}(d, \mathcal E)$ of $d$-planes associated to a vector bundle $\mathcal E$ on a scheme $X$ is by definition a fibre bundle whose fibre at each point $x \in X$ is isomorphic to the Grassmann variety $\mathbb G(d, \mathcal E\otimes k(x))$ of $d$-planes in the vector space $\mathcal E\otimes k(x)$.
If $X$ is projective over a field and $\mathcal E$ is very ample, then $\mathbb G_{X}(d, \mathcal E)$ is naturally embedded into a projective space with relative Pl\"ucker embedding over $X$. In this talk I explain a degree formula for $\mathbb G_{X}(d, \mathcal E)$ (jointwork with T. Terasoma) and its applications.
Simone Marchesi (IMECC-UNICAMP）
Title：Uniform Steiner bundles
Abstract:One of the most classical problems in vector bundles is the study and classification of uniform bundles, i.e. vector bundles whose splitting type is constant for any chosen line of the base projective space. In this talk we will recall the main results concerning this topic and we will introduce a new technique to construct families of uniform Steiner bundles which are not homogeneous. After proving upper and lower bounds for the rank of uniform Steiner bundles, a conjecture will be proposed of what we strongly believe is the global picture for uniform Steiner bundles. This is a joint work with Rosa Maria Mir\'o-Roig (UB-Spain).
Chikashi Miyazaki （Kumamoto University）
Title:Cohomological Criteria for Vector Bundles on Multiprojective Space
Abstract: This talk is concerned with cohomological property of vector bundles towards Horrocks-type criteria on multiprojective space. The Horrocks theorem says that a vector bundle $E$ on ${\Bbb P}^n$ is a direct sum of line
bundles if and only if ${\rm H}^i(E(t))=0$ for all $1 \le i \le n-1$ and $t$, in other words, any maximal Cohen-Macaulay module over a regular localring is free. A Buchsbaum vector bundle on ${\Bbb P}^n$ is also known to be isomorphic to a direct sum of line bundles and sheaves $\Omega^p_{{\Bbb P}^n}(\ell)$ of differential forms with some twist by Chang and Goto. We will study the corresponding behaviour of vector bundles on multiprojective space by using the syzygy technique including the Castelnuovo-Mumford regularity.
Joan Pons-Llopis (University of l'Aquila)
Title: The Minimal Resolution Conjecture (MRC) for points on the cubic scroll
Abstract: The MRC (as it was formulated by Lorenzini in the case of the projective space and by Mustata for a general variety) claims that the Betti numbers of the minimal graded resolution of a general set of points on a projective variety are as small as possible. This conjecture, in the two dimensional case, has been proven for the projective plane, the quadric and the general cubic and quartic surfaces.
The goal of this talk is to explain our approach using techniques of liaison theory (already exploited in other cases) to prove MRC for certain cardinalities of points on cubic scrolls.
Eiichi Sato（Kyushu University）
Title: On Fano bundles of higher rank on the projective spaces
Abstract: We study the structure of Fano bundles of higher rank, mainly, on the projective space. It is well-known that there are many results about rank-2 Fano bundles on smooth projective varieties and in particular that almost of them split on $ n (\geq 3 )$-dimensional projective space. The latter is deeply connected with Harthorne conjecture. On the other hand it seems to us that there are very few results of higher rank. In this talk we consider them on projective spaces and hyperquadrics.
Jean Valles (University of Pau and Pays de l'Adour)
Title: Unexpected divisors of jumping lines
Abstract: Let $E$ a stable rank two vector bundle over $\mathbb{P}^n$. When $c_1(E)$ is odd, the expected codimension in $\mathrm{G}(1,\mathbb{P}^n)$ of the scheme of jumping lines of $E$ is $2$. When $c_1(E)$ is even, the expected codimension of the scheme of jumping lines of order $\ge 2$ of $E$ is $3$. These codimensions are expected, but does it exist a stable rank two vector bundle with an unexpected divisor of jumping lines?
In $\mathbb{P}^2$, the answer is yes and there are, as far as I know, only two kinds of divisors: lines and smooth conics. When the divisor is supported by a single conic the bundle is the so-called Schwarzenberger's bundle.
In $\mathbb{P}^3$, Gruson and Peskine assert (their proof was improved by Han) that an unexpected irreducible divisor $K\subset \mathrm{G}(1,\mathbb{P}^n)$ of jumping lines is always the set oflines meeting an irreducible space curve $C$ where $C$ is the zero locus of a special section.
This nice result probably deserves to be better known. Many related problems are still open: does an irreducible unexpected cubic curve in $\mathbb{P}^2$ exist,
what happens for bundles of higher rank?
Kenta Watanabe (Nihon University)
Title: On the classification of rank two ACM bundles on quartic hypersurfaces in $\mathbb{P}^3$
Abstract: Let $X$ be an algebraic surface, and $L$ be a very ample line bundle on $X$. Then we call a vector bundle $E$ on $X$ an Arithmetically Cohen-Macaulay (ACM for short) bundle with respect to $L$ if $H^1(X,E\otimes L^{\otimes l})=0$, for any integer $l$. It is interesting to investigate the existence of indecomposable ACM bundles on $X$ of higher rank with given Chern classes and give a classification of them. However, it is difficult to do them, even if $X$ is a hypersurface in $\mathbb{P}^3$.
In this talk, we consider ACM bundles of rank two on quartic hypersurface in $\mathbb{P}^3$. Recently, Gianfranco Casnati has classified rank two ACM bundles on general determinantal quartic hypersurfaces in $\mathbb{P}^3$, by the Chern classes and the zero locus of sections of them. I will recall his work and introduce my recent work about the classification of ACM bundles on quartic hypersurfaces in $\mathbb{P}^3$.
Organizers:
Takeshi Abe (Kumamoto)
Takuro Abe (Kyushu)
Chikashi Miyazaki (Kumamoto)
This conference is supported by JSPS KAKENHI Grant number 16H03924 (Takuro Abe)