# 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 *s _{1}(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 *f *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

*D*-Lie algebras. Associated to the D-Lie algebra F(L,f):=L(f) I will construct

*U(L(f))*with the property that there is an exact equivalence of categories

*U(L(f))*to construct Ext and Tor groups of

*L*and

*L(f).*

*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 *H ^{6} *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.