Seminarium Geometria Algebraiczna

We will present the paper by D. Inoue under the same title. In the paper the author presents constructions of derived equivalent and non-birational pairs of Calabi-Yau manifolds obtained as sections of projective joins between some del Pezzo manifolds. We shall discuss several example described in the paper. In particular, we will see candidate mirror families admitting several points of maximal unipotent monodromy, where each of these points corresponds via mirror symmetry to a different Calabi-Yau manifold.

Przedstawię konstrukcję grup Chow $CH_k (X)$ rozmaitości quasi-rzutowej $X$. Są one ilorazami grupy $k$-cykli $Z_k (X)$ - wolnej grupy abelowej generowanej przez domknięte $k$-wymiarowe podrozmaitości $X$ - przez relację wymiernej równoważności, analogicznej do relacji liniowej równoważności dywizorów. Na koniec udowodnię lemat o lokalizacji.

Na podstawie: C. Voisin, Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Annals of Mathematics Studies, 187. Princeton University Press.

Przedstawię własności funktorialne grup Chow. Pokaże jak grupy CHow zachowują się przy morfizmach właściwych oraz płaskich. Udowodnię że morfizm właściwy zadaje operację "push-forward" natmiast morfizm płaski operację "pull-back" cykli algebraicznych. Referat zakończę dowodem formuły rzutowania.

A $\mu$-stable vector bundle $\mathcal E$ of rank 2 with $c_1(\mathcal E) = 0$ on $\mathbb P^3 (\mathbb C)$ is called mathematical instanton bundle if $H^1(\mathbb P^3 , \mathcal E(−2)) = 0$. In this talk, we will study the definiton of mathematical instanton bundles on Fano 3-folds and the construction of them on the one which is the blow-up of $\mathbb P^3$ at a point. This talk is based on the joint work with Gianfranco Casnati, Emre Coskun and Francesco Malaspina

We will present a notion of a Jacobian of a curve and its generalisation called an intermediate Jacobian of a variety. We will present some of its properties, especially the universal property that leads to a notion of an Albanese variety. We will construct an Abel-Jacobi map and show some applications of the above notions. The talk is based on a book by Claire Voisin entitled Hodge Theory and Complex Algebraic Geometry I

Given a finite morphism between smooth curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. It is a classical result that the Prym map is generically injective for étale double coverings. In this talk we will give an introduction to the Prym maps. We will then consider the Prym map between the moduli space $\mathcal{R}_{g,r}$ of double coverings over a genus $g$ curve ramified at $r$ points, and $\mathcal{A}^{\delta}_{g-1+r/2}$ the moduli space of polarized abelian varieties of dimension $g-1+r/2$ with polarisation of type $\delta$. We will show a constructive proof of the generic injectivity of the Prym map when $r \geq max{6, 2/3 (g+2)}$. This a joint work with J.C. Naranjo.

Based on the paper "K3 surfaces with maximal finite automorphism groups containing $M_20$" by C. Bonnafé and A. Sarti, we will present the results describing the finite groups of automorphism acting faithfully on K3 surfaces and containing the Mathieu group $M_20$ (the biggest possible finite group acting faithfully and symplectically on K3's). This will be preceded by a brief introduction to K3 surfaces.

A vector bundle $\mathcal{E}$ on a projective variety $X$ in $\mathbb{P}^{N}$ is Ulrich if $H^*(X,\mathcal{E}(-k))$ vanishes for $1 \leq k \leq dim(X)$. It has been conjectured by Eisenbud and Schreyer that every projective variety carries an Ulrich bundle. Even though this conjecture has not been proved or disproved, another interesting question is worth considering: classify projective varieties as Ulrich finite, tame or wild type with respect to families of Ulrich bundles that they support. In this talk, we will show that this trichotomy is exhaustive for certain del Pezzo surfaces with any given polarization. This talk is based on a joint work with Emre Coskun.

We shall construct arbitrary dimensional Zariski Calabi-Yau varieties in odd characteristics such that p is not equivalent to 1 modulo 12.

Online seminar, join Zoom Meeting https://zoom.us/j/96567747460?pwd=eGdlTXF4UVJrYW40cjdwc2lhTDdOQT09 Meeting ID: 965 6774 7460 For password email Paweł Borówka