![]() Exceptional younger students and interested graduate students will also be considered.īy whom: Lectures will be given by senior PhD students, postdocs, but also by some senior researchers. All lectures will be given in English, so we also welcome students from anywhere abroad. The school is not limited to lectures, but will also include daily discussion sessions, problem solving and independent work.įor whom: The school is intended for senior undergraduate and early graduate (MSc) students mainly from the Balkans region/Southeast Europe. The aim: The school aims to provide an intensive and thorough course on a broad range of astrophysical topics. This school will provide a theoretical foundation for various astrophysical topics that will allow us to understand the science cases of the probes from the solar system to cosmological scales. ![]() A plethora of new observational instruments have either been recently launched or are planned to be launched soon (LIGO-Virgo-KAGRA, EHT, DESI, Euclid, DKIST). This year we are starting the 3rd cycle of PSI summer schools, with the focus on Astrophysics. The school is supported by International School for Advanced Studies (SISSA, Trieste). PSI is organized every summer in Petnica Science Center (Valjevo, Serbia) and it covers a wide range of topics in theoretical physics and astrophysics. Petnica Summer Institute (PSI) aims to provide lectures to undergraduate and early graduate students by senior PhD students, young postdocs and researchers. We are happy to present Petnica Summer Institute 2023 – Summer School on Theoretical Astrophysics. Hartshorne, "Algebraic geometry", Springer (1977) MR04631.More info: external link Date: - Contact: psipetnica.rs Location: Petnica, Serbia Zariski, "The theorem of Bertini on the variable singular points of a linear system of varieties" Trans. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" Mem. Akizuki, "Theorems of Bertini on linear systems" J. Bertini, "Introduction to the projective geometry of hyperspaces", Messina (1923) (In Italian) The Bertini theorems apply to linear systems of hyperplane sections, without restrictions on the characteristic of the field. If the characteristic of $k$ is finite, the corresponding theorem is true if the extension $k(V)/k(W)$ is separable. If $\dim W = 1$, Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping $\phi-L : V \to W$ are irreducible and reduced if the function field $k(W)$ is algebraically closed in the field $k(V)$ under the imbedding $\phi_L^* : k(W) \to k(V)$. all except a closed subset in the parameter space $P(L)$ not equal to $P(L)$) are irreducible reduced algebraic varieties.Ģ) Almost all divisors of $L$ have no singular points outside the basis points of the linear system $L$ and the singular points of the variety $V$>.īoth Bertini theorems are invalid if the characteristic of the field is non-zero.Ĭonditions under which Bertini's theorems are valid for the case of a finite characteristic of the field have been studied. The following two theorems are known as the first and the second Bertini theorem, respectively.ġ) If $\dim W > 1$, then almost all the divisors of the linear system $L$ (i.e. Let $V$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0, let $L$ be a linear system without fixed components on $V$ and let $W$ be the image of the variety $V$ under the mapping given by $L$. ![]() Two theorems concerning the properties of linear systems on algebraic varieties, due to E.
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