Seminarium Geometria Algebraiczna

The pro-\'etale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual \'etale fundamental group defined in SGA1 and the more general group introduced in SGA3. It controls local systems in the pro-\'etale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite \'etale covers. I will explain the generalization of some foundational results of Grothendieck from the \'etale to the pro-\'etale fundamental group. Most notably, the homotopy exact sequence over a field and over a general base scheme. To prove the first statement, we study Galois actions on free topological products of groups. To prove the second one, we construct an "infinite" (i.e. non-quasi-compact) analogue of Stein factorization. On the way, we will mention a general van Kampen theorem and the K\"unneth formula for the pro-\'etale fundamental group. If time allows, I will discuss existence of the specialization morphism for the pro-\'etale fundamental group.

The talk is devoted to the geometry of elliptic normal curves. Firstly, we will show how to embed an elliptic curve defined over the field of complex numbers as a linearly normal curve of degree n. We will focus on the case $n=6$ and determine the basis of the space of quadric hypersurfaces through an elliptic normal curve $C_6$. Then we will prove some results about generators of ideals of images of $C_6$ under projections from a general point and a general line.

In the talk I will present selected episodes from the history of Intersection Theory. We will discuss a few problems and methods illustrating some of the ideas which will be of most concern to us in the series of talks based on Fulton, Introduction to Intersection Theory in Algebraic Geometry.

We continue the discussion of Introduction to Intersection Theory in Algebraic Geometry by W.Fulton. In this talk we cover the material from the second chapter Multiplicity and normal cone.

In the talk I will further generalize the notion of intersection product. I will start with a few basic notions from the theory of cohomology and Chern classes. Later I will define rational equivalence relation for algebraic cycles and finally I will use these notions to get the intersection product of a Cartier divisor and an algebraic cycle. The talk will be based on the third chapter of the book: Introduction to intersection in algebraic geometry by W. Fulton.

Let $f\in S_k(\Gamma_0(N))$ be a weight $k$ normalized newform with rational coefficients. There exists two complex numbers $u^+$ and $u^-$ such that $(2\pi i)^{-m}L(f,m)/u^+\in \mathbb Q$ (resp. $(2\pi i)^{-m}L(f,m)/u^-\in \mathbb Q$) for $m\in\{1,\dots,k-1\}$ even (resp. odd). I will present a result of Shimura which gives a similar formula for special $L$-values of twists of $f$ by a Dirichlet character ([G. Shimura, On the periods of modular forms. Math. Ann. 229 (1977), no. 3, 211–221]).

I will explain the proofs of the main theorems presented last time. I will start by discussing a counterexample to the statement that every geometric covering of $X_{\bar{k}}$ is defined over a finite field extension of $k$ and will explain how to gain insights from this example to prove the fundamental exact sequence for $\pi_1^{\mathrm{proet}}$. I will also mention how to use the pro-\'etale topology to define an "infinite" analogue of Stein factorization in this setting. If time allows, I will mention the problem of existence a specialization morphism for the pro-etale fundamental group. The specialization morphism for the étale fundamental groups of Grothendieck cannot be generalized word-for-word to the more general pro-\'etale fundamental group of Bhatt and Scholze. I will discuss a counterexample to this statement. It turns out, however, that one can deal with this problem by applying a rigid-geometric point of view.

Using Chern and Segre classes introduced in the first part of this talk I will present the intersection formula $X\cdot V = \{c(\mathcal N)\cap s(W,V)\}_{n-d}$.

In my talk I will define the Gysin homomorphism and state its basic properties. Then I will introduce intersection ring of a non-singular variety (also known as the Chow ring), together with examples and applications. The talk will be based on the fifth chapter of the book Introduction to intersection in algebraic geometry by W. Fulton.

Continuing the series based on Introduction to intersection in algebraic geometry by W. Fulton, I will focus on chapters 6th and 7th dealing with the degeneracy loci of homomorphisms between vector bundles and refinements.

Continuing the series based on Introduction to intersection in algebraic geometry by W. Fulton, I will focus on chapters 6th and 7th dealing with the degeneracy loci of homomorphisms between vector bundles and refinements.

In the last talk of the series of talks based on the book Introduction to intersection in algebraic geometry by W. Fulton I will present the Riemann-Roch theorem.

The talk will concern topics from real algebraic geometry. We will start the talk by defining flexible ringed spaces and showing some simple properties like a version of Nullstellensatz. Then we will give the definition of $k$-regulous functions and prove that $\mathbb R^n$ equipped with the sheaf of them is an example of such a flexible ringed space by use of some semialgebraic geometry and introduction of constructible topology on $\mathbb R^n$.

The aim of this seminar will be to characterise primes of the form $x^2 + ny^2$. I will begin with preliminary information about fields and Galois extensions. Then I will introduce some essential concepts from Class Field theory like finite and infinite primes, Artin symbol or ring class field. Then, we will prove the main theorem which characterises such primes. At the end, I will talk about how it is related to the j-invariant of some elliptic curve.

We say that a set $X$ consisting of linear subspaces of \mathbb{P}^n has good postulation in degree $t$, if the vanishing along $X$ imposes independent conditions on forms of degree $t$. During my talk I will discuss the theorem saying that the set of general planes in $\mathbb{P}^4$ has good postulation in degrees $t\le11$. I will focus on the most complicated case analysed in the proof (20 planes in degree 11) and some computer methods that could lead to the proof of good postulation in some higher degree cases.