Seminarium Geometria Algebraiczna

I will present some algorithms for computation of arithmetic subgroups of groups related to algebraic geometry developed in series of papers by A.S.Detinko, D.L.Flannery and E.A. O’Brien, and A.S. Detinko,D.L. Flannery, A. Hulpke. I will start with some motivation: arithmetic groups in $\operatorname{SL}_n(\mathbb Z)$ related to the level structures for elliptic curves and monodromy groups of the Picard-Fuchs differential equations for the period integrals. I will briefly recall the basic definition and then discuss the example of fourteen $CY(3)$ operators: seven with arithmetic and seven with thin monodromy groups.

A $\mu$-stable vector bundle $\mathcal{E}$ of rank 2 with $c_1 (\mathcal{E})=0$ on $\mathbb{P}_{\mathbb{C}}^{3}$ is called a mathematical instanton bundle if $\mathrm{H}^1 (\mathbb{P}^{3}, \mathcal{E}(-2))=0$. This type of bundle has been generalized to other varieties in various ways. First, it has been generalized to odd-dimensional projective spaces by M. M. Capria and S. M. Salamon, then to non-locally free sheaves of any rank on arbitrary projective spaces by M. Jardim. Then, D. Faenzi and A. Kuznetsov extended the definition to other Fano threefolds, and later, V. Antonelli and F. Malaspina modified the definition to apply to any polarization of Fano threefolds, introducing the concept of an $h$-instanton bundle. Finally, V. Antonelli and G. Casnati further broadened the definition to cover any polarized variety $(X,h)$. In this talk, we will focus on rank 2 $h$-instanton sheaves on ruled Fano threefolds with Picard rank 2 and index 1. This is a joint work with Marcos Jardim.

Kontsevich suggested that the Landau—Ginzburg models provide a good formalism for investigations around the mathematical mirror symmetry problem. A different perspective on Landau-Ginzburg models is discussed in light of this claim. As a result, certain results of Abouzaid-Auroux-Katzarkov can be recovered differently. By using this different angle, we can make new advances regarding a conjecture of Kontsevich--Soibelman on a version of the Strominger-Yau-Zaslow mirror problem

One form of Serre duality states that for an n-dimensional variety over a field its i-th and (n-i)-th cohomology groups are dual to each other as vector spaces. Verdier generalised analogous result in algebraic topology - Poincare duality - using the language of derived categories, essentially proving that the derived direct image functor acting on sheaves on locally compact spaces admits a right adjoint. This talk is devoted to a proof of a result by Neeman, called Grothendieck duality, which in the same spirit generalizes Serre duality to quasi-compact quasi-separated schemes, and which will be proved in the context of infinity-categories. First, I will briefly introduce the necessary notions from infinity-categories and present a proof of Brown representability theorem. In the second part of the talk, we will show that the category of quasicoherent sheaves on a qcqs scheme satisfies the hypothesis of Brown's theorem.

A Nikulin orbifold $Y$ is obtained by partial resolution of the quotient of a $K3^{[2]}$-type manifold $X$ by a symplectic involution $i$: any automorphism $s$ of $X$ that commutes with $i$ induces therefore an automorphism on $Y$. I will describe induced symplectic involutions on Nikulin orbifolds, starting from the action of groups of order 4 on $K3^{[2]}$-type manifolds, and present some classification results.

In this talk, we will discuss the fundamental concepts and constructions of Shimura curves via Fuchsian groups. We will then explore how these curves provide a moduli space for certain special abelian varieties. The lecture is based on Three Lectures on Shimura Curves by John Voight

The goal of this talk is to present classification Fano 3-folds. I will start with a short discussion of del Pezzo surfaces.

Given a finite morphism between smooth curves one can canonically associate to it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. Classically, the Prym map has been studied for cyclic covers, where its generic injectivity has been established in many cases. In this talk we will go beyond cyclic covers and focus on unramified Galois covers of genus two curves with the Galois group isomorphic to the product of two cyclic 3-groups. In particular, we will show that the associated Prym map is injective. We will start with a brief introduction to complex abelian varieties and the Prym theory. This is a joint work with Paweł Borówka.

We are going to speak about V.D.Goppa's construction of error-correcting codes using algebraic function. An error-correcting code is a subspace of $\mathbb F (q,n)$ - the $n$-dimensional standart vector space over a finite field $\mathbb F(q)$. Such codes are in widespread use for the reliable transmission of information. As observed V.D.Goppa in 1975, one can use algebraic function fields over $\mathbb F(q)$ to construct a large clas of interesting codes. Properties of these codes are closely related to properties of the corresponding function field, and the Riemann-Roch Theorem provides estimates, sharp in many cases, for their main properties. So we start with some basic results of the theory of algebraic function fields (valuation, places, divisors and so on) and some main concepts of coding theory. We consider (shortly) Reed-Solomon codes over $\mathbb F(q)$, as algebraic geometrical codes (AG-codes) are a very natural generalisation of Reed-Solomon codes. Then we continue discussion about AG-codes and their main properties. At the end we consider the codes constructed by means of rational function field.

Literature: H. Stichtenoth Algebraic Function Fields and Codes

The relationship between convex geometry and intersection theory has deep historical roots, tracing back to classical results in enumerative geometry. Our motivations come from two interconnected problems regarding projections of geometric objects in four-dimensional spaces:

1. Let $A$ be a compact convex set in $\mathbb{R}^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the areas of the six coordinate projections of $A$ to $\mathbb{R}^2$. Which sextuples of nonnegative real numbers can occur in this way?
2. Let $S$ be an irreducible surface in $(\mathbb{P}^1)^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the degrees of the six coordinate projections from $S$ to $(\mathbb{P}^1)^2$. Which sextuples of nonnegative integers can occur in this way?

These questions find their solutions encoded in the Plücker relations for the Grassmannian $\text{Gr}(2,4)$ over a triangular hyperfield. The talk is based on joint work with Daoji Huang, June Huh and Botong Wang.

During this talk we will apply results from the previous talk to some Calabi--Yau threefolds, in particular the Fermat quintic threefold and Calabi--Yau (crepant) resolutions of quotients of products of three elliptic curves with a particular group action.

The talk will be based on the paper C. Borcea: Calabi--Yau Threefolds and Complex Multiplication.

Consider a degeneration of Kähler manifolds over a punctured disk whose central fiber is snc. The A'Campo space (constructed jointly with J. F. de Bobadilla) extends it to a symplectic fibration over an annulus, whose restriction to the inner ("radius zero") circle has a simple combinatorial structure. In fact, it is decomposed to pieces which are isotropic torus bundles over the strata of the central fiber, and the monodromy acts as translation in these tori.

In my talk I will survey this construction and highlight its two applications. First, given the explicit description of the monodromy one can compute its Floer homology using McLean's spectral sequence, and thus infer a positive answer to the Zariski multiplicity conjecture for μ-constant families of isolated hypersurface singularities. Second, the Lagrangian tori over the $0$-dimensional strata have very similar properties as those expected by the Kontsevich-Soibelman conjecture in the context of maximal Calabi-Yau degenerations: for instance, they fill all the volume of the fiber, are asymptotically special, and collapse in the Gromov-Hausdorff limit.

I will present van Straten's Conjecture about the monodromy matrices in the scaled Frobenius base of solutions of an $\operatorname{CY}(3)$ differential operator and its generalizations (gamma class). Finally, I will discuss the gamma class for several Calabi-Yau operators with several MUM points.