I will present an example of a one-parameter family of Calabi-Yau threefolds $X_t$ such that

• local monodromy at $0$ is trivial,
• a generic fiber is a smooth,
• the special fiber $X_0$ is reducible: one component is a smooth, rigid Calabi-Yau threefold, the other is a conic bundle.

There is no analogous example in dimension 2 because a well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family.

Joint work with D. van Straten (Johannes-Gutenberg Universitaet Mainz): preprint arXiv:1812.01622 [math.AG]

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.

I will describe details of the construction of conjectural mirror families of sections of joins between some del Pezzo manifolds (given in the paper "Calabi--Yau 3-folds from projective joins of del Pezzo manifolds" by D. Inoue). Candidate mirros are constructed as desingularized fiber products of some rational elliptic fibrations (this contruction was introduced by C. Schoen). I will discuss details of Schoen's family and the relation between fiber products of varieties and Hadamard product of period integrals.

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.

I will give a brief introduction to intersection theory. My main goal will be establishing the projection formula $p_*(p^*Z\cdot Z')=Z\cdot p_*Z'$ for a proper morphism $p$. The talk will be based on: Claire Voisin, Hodge Theory and Complex Algebraic Geometry II and William Fulton, Intersection Theory.

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.

In my talk I will define the notion of a correspondence between two smooth varieties and introduce the categories of effective motives and Chow motives. Then I will present a few examples. The talk will be based on: C. Voisin, Chow Rings, Decomposition of the Diagonal, and the Topology of Families.

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

I will talk about different strategies for compactifying moduli spaces, including GIT, MMP, SBB, etc. Hopefully some of my work on moduli of CY’s or linear subspace arrangements will be sketched.

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

Podam przykład płaskiej rozmatości wymiaru 48, której reprezentacja holonomii ma wlasność taką że jej wszytkie $\mathbb R$-nieprzywiedlne skladniki są typu kwaternionowego. Wszytkie powyższe pojecia zostaną zdefiniowane i omówione na początku wykładu.

Wspólna praca z G. Hissem i R. Lutowskim.

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.

Przedstawię szkic dowodu prawdziwości Hipotezy Voisin dla pewnych gładkich rozmaitości rzutowych wymiaru co najwyżej 5 (twierdzenie 4.1 z pracy Bini, Laterveer, Paceinza, Voisin's Conjecture for zero-cycles on Calabi-Yau varieties and their mirrors ). Omówię najważniejsze wykorzystane w dowodzie pojęcia, w tym m.in. motywy skończenie wymiarowe oraz filtracje niveau oraz coniveau.

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 the direct image of $\mathcal{E}$ is trivial for some linear projections $X$ into $\mathbb{P}^{n-1}$. It was conjectured that on any variety there exist Ulrich bundles. In this talk, we will study the construction of stable Ulrich bundles of rank 1 and 2 on Fano 3-folds which are blow-ups of $\mathbb{P}^{3}$ along genus 3, degree 6 curves.

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.

The main purpose of my talk is to discuss the following question: what is the maximum possible number $t$ of triple points for an irreducible rational plane curve of a given degree $d$? The answer is known for small degrees, for example $t=1$ for $d=4$ and $t=2$ for $d=6$. During my talk, I am going to present some results and examples of such curves for small degrees. I am also going to introduce some computer-assisted methods that can be used in order to find curves of degree $d$ with a given number of triple points. The talk is based on my joined work with Joaquim Roe.

Online seminar on MS teams

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