# Foredragsholdere, Nasjonalt algebramøte 2019

Georg Hitcing: Segre invariants of bundles over curves, and inflectional loci of scrolls

Abstract: Let E be a vector bundle over a smooth curve C, and $$S=\mathbb{P}(E)$$  the associated projective bundle. We describe the inflectional loci of certain projective models of $$f:S\rightarrow \mathbb{P}^n$$i terms of invertible subsheaves of E. This gives a geometric characterisation of the Segre invariant s1(E), which leads to new geometric criteria for semistability and cohomological stability of bundles over C. We sketch how these ideas can be used to show that for general enough S and f, the inflectional loci are all of the expected dimension.

Helge Maakestad: D-Lie algebras, extensions and cohomology of connections.

Abstract: Given an A/k-Lie-Rinehart algebra L, we may consider the category Conn(L) of L-connections and maps of L-connections. A fundamental problem in the theory of connections, is to give an explicit description of Conn(L) as a module category over an associative ring, and in this talk I will solve this problem.

An L-connection $$(E,\nabla)$$ has "curvature type f" where is a 2-cocycle for L with values in A, if for any element $$e\in E, x,y \in L$$ it follows $$R_{\nabla}(x,y)(e)=f(x,y)e$$. If $$f\neq0$$ it follows $$\nabla$$ is a non-flat connection. In a previous paper (J. Algebra 2015) i studied the category of connections of curvature type f and introduced Ext and Tor groups of such connections, generalizing the classical case when f=0 of a flat connection. In this lecture I will generalize this further to the case of an arbitrary connection $$\nabla \in Conn(L)$$ with no condition on the curvature. I will introduce a new structure - a D-Lie algebra - and construct for any Lie-Rinehart algebra L, and any 2-cocycle f, a functor F:=F(L,f) from the category of A/k-Lie-Rinehart algebras

to the category of D-Lie algebras. Associated to the D-Lie algebra F(L,f):=L(f) I will construct
an enveloping algebra U(L(f)) with the property that there is an exact equivalence of categories
$$\psi_f: Conn(L) \rightarrow Mod(U(L(f))$$
preserving injective and projective objects. I will use the enveloping algebra U(L(f)) to construct Ext and Tor groups of L and L(f).
Questions that can be asked for connections on Lie-Rinehart algebras may be asked for connections on D-Lie algebras, and I will discuss extensions and non-abelian extensions of D-Lie algebras if there is time available.

Marco Rampazzo, UiS: Derived equivalence of Kanemitsu roofs: the case of K3 surfaces of degree 12

Abstract: We construct a family of K3 surfaces of degree 12 given by zero loci of sections of Ottaviani bundles on a five dimensional quadric. We prove that the generic pair of such varieties is derived equivalent but not isomorphic by means of an explicit embedding of the middle cohomology in the H6 of a Fano 6-fold. The derived equivalence lifts to an equivalence of matrix factorizations categories, connecting two Landau–Ginzburg models despite the lack of a GIT phase transition. This example is an instance of a more general construction related to the roofs introduced by Kanemitsu.

Bjørn Skauli, UiO: Curve classes on Calabi-Yau Hypersurfaces in Toric Varieties

Abstract: The Integral Hodge Conjecture is a question about the relation between the topological and algebraic structure of smooth complex varieties. It asks whether for a variety X certain cohomology classes, the integral Hodge classes, are generated by the classes of algebraic subvarieties of X. This is false in general, but by imposing restrictions on the variety X, one can obtain positive results. An especially interesting case is whether the degree 2n-2 integral Hodge classes are generated by algebraic curves contained in X. For Calabi-Yau threefolds, this is proven by Voisin. In this talk, we will present the Integral Hodge Conjecture and prove the Integral Hodge Conjecture for degree 2n-2 classes for Calabi-Yau varieties of dimension three or greater, constructed as anticanonical hypersurfaces in smooth toric varieties. The proof relies on a result by Casagrande on curve classes in smooth toric varieties and the toric Minimal Model Program.

Tuyen Trung Truong, UiO: An algorithm for bounded birationality problem

Abstract:  In this talk I will present an algorithm to check whether, given two varieties X and Y over an algebraically closed field K and a number d, there is a birational map f:X-->Y of degree bounded by d. The same method can be applied to bounded embedding: whether there is a closed embedding from X into Y. Note that if there is no bound on the degree, then by result of Kanel-Belekov and Chilikov (in comments to Question 1.54 in the following link http://aimpl.org/rationalityag/1/), there is no such algorithm for X the affine space of dimension 11 in general in the closed embedding question. For future work: I am looking to simplify the algorithm to apply to interesting and specific cases.

Publisert 13. sep. 2019 13:26 - Sist endret 4. nov. 2019 11:13