Seminar Algebraic Geometry

I will present Tian-Todorov unobstructedness theorem for Calabi-Yau manifolds, I will recall necessary information about deformations of complex manifolds
zoom seminar: https://zoom.us/j/94487954519?pwd=TXZFSy8yY1dTUGo1MSt5VGZpRnMxQT09

I will present a counter example to Tian-Todorov unobstructedness theorem for Calabi-Yau manifolds in positive characteristic, I will also present basic properties of nodal quintic hypersurfaces in $\mathbb P^4$ (topology of a small resolution, defect, cubic form on the Picard group) and give examples with highest numbers of nodes.
ms teams online seminar

W tym wykładzie omówię teorię rezolwent wolnych nad pierścieniami Noetherowskimi. Wychodząc od Twierdzenia Hilberta o syzygiach, omówię prace Buchsbauma, Eisenbuda i Hochstera z lat 70-tych i opowiem o ostatnich rezultatach rozwijajacych te teorie.

In this talk I will present basic notions used in the study of the minimal model problem. I will start from Kleiman's criterion for ampleness and then define terminal, canonical and log-terminal singularities.
The talk will be based on: Kawamata, Matsuda and Matsuki Introduction to the Minimal Model Problem, sections 0-1 and 0-2.

My talk is the second one in the series of talks based on the Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki. I will begin with a short statement of the fundamental conjectures of the Minimal Model Program and then I will move on to the topics related to the canonical varieties, including Iitaka dimension, Kodaira's lemma and the notion of a variety of general type. The third part of my talk will focus on the statement and the proof of the Covering Lemma.

In the first part of my talk, I'll focus on the proof of Rationality Theorem. Then I will illustrate how Rationality Theorem can be used to prove Cone Theorem, a structure theorem on the closed cone of curves of an algebraic variety. This theorem is one of the key steps toward the theory of minimal models.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

We continue the discussion of the Minimal Model Program with description of types of contractions of extremal rays, namely contractions of fiber type, contractions of divisorial type and contractions of flipping type.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

We continue the discussion of the MMP by giving generalizations of Non-Vanishing Theorem and the Base Free Theorem replacing the ample (or nef and big divisors) in the original statements by nef and abundant divisors.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

This will be the presentation of my bachelor thesis. The aim of this seminar will be to present proof that the j-invariant of an elliptic curve with complex multiplication is an algebraic integer. I will begin with preliminary information including basic facts about imaginary quadratic fields and elliptic curves with complex multiplication and then I will move on to the topics related to the modular forms and modular polynomials. The third part of my talk will focus on the proof of the main theorem. I will conclude with a nice application of this theory to justify why Ramanujan Constant is almost integer.

Good semigroups are defined as a family of submonoids of $N^d$, properly containing the family of value semigroups of rings of curves singularities. In the case $n=1$, they are just numerical semigroups and their properties, in relation with those of the associated monomial curves have been well studied. Extending those results to the general case, we construct and study the Apéry sets of good semigroups and, focusing mostly on the case $n=2$, we show how these sets are used to describe properties such as symmetry and almost-symmetry. In the case of value semigroups of curves, such properties correspond to Gorensteiness and almost-Gorensteiness of the associated rings.

Analogously to the work which followed the classification of finite symplectic actions on K3 surfaces by S. Mukai in 1988, we will use the paper Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$ by G. Höhn and G. Mason to study some very symmetric $K3^{[2]}$ type fourfolds, i.e ones acted upon via (not necessarily symplectic) automorphisms by the largest possible groups strictly containing the maximal groups from the article.

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. The function $p_\mathcal{A}(n,k)$ counts the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m})_{n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$, and apply them to the special cases of $\mathcal{A}$. In particular, for a fixed parameter $k$, we investigate both the upper bound for the odd density of $p_\mathcal{A}(n,k)$ and the lower bound for the density of $\{n\in\mathbb{N}: p_\mathcal{A}(n,k)\not\equiv0 \pmod{m}\}$. Furthermore, we present some new results related to restricted $m$-ary partitions and state a few open problems at the end of the talk.

We present an algorithm for counting points on a double octic Calabi-Yau threefold associated to a configuration of eight planes over a finite field based on the existence of an elliptic curve fibration.

This paper contains results of PhD studies interrupted by sudden death of Aleksander Czarnecki on February 28, 201.

aleksander.czarnecki.im.uj.edu.pl

Given a set $X$ consisting of subspaces of $\mathbb{P}^n$ one can ask whether vanishing along $X$ imposes independent conditions on forms of degree $t$. If this is the case, we say that $X$ has good postulation in degree $t$. By the well-known Hartshorne-Hirshowitz Theorem, a set of general lines in $\mathbb{P}^3$ has good postulation in any degree. That result leads to the study of the postulation of general planes in $\mathbb{P}^4$. In my talk I will present the method to show that the set of general planes has good postulation in certain degrees.