Contributed Talks Schedule
Week 1
Monday 7/14 - NATRS 113:
Chair: Andrés Jaramillo Puentes
1:30 Giulio Codogni
2:05 Osamu Fujino
2:50 Andrea Fanelli
3:25 Jack Hall
4:10 David Swinarski
4:45 Piotr Pokora
Tuesday 7/15 - NATRS 113:
Chair: Mark Shoemaker
1:30 Dmitry Zakharov
2:05 Anoop Singh
2:50 Yongnam Lee
3:25 Taro Sano
4:10 Stephen Coughlan
4:45 Uriya First
Thursday 7/17 - NATRS 113:
Chair: Seth Ireland
1:30 Jianqi Liu
2:05 Luca Tasin
2:50 Ishan Banerjee
3:25 Valentina Beorchia
4:10 Tomasz Szemberg
Friday 7/18 - NATRS 113:
Chair: David Swinarski
1:30 Taro Fujisawa
2:05 Luca Schaffler
2:50 Changho Han
3:25 Jayan Mukherjee
Titles and Abstracts
Giulio Codogni. Slope inequalities, effective positivity results for Hodge bundles and applications
We will explain various higher dimensional generalizations of the Xiao-Cornalba-Harris slope inequality, and we will present several effective positivity results for Hodge bundles. We will talk about applications regarding the volume of integrable foliations, the volume and the ample cone of KSB moduli spaces, and the cardinality of the auotmorphism group of KSB fibrations. This is based on joint works with Zs. Patakfalvi, L. Tasin, F. Viviani.
Osamu Fujino. Minimal model theory for projective morphisms between complex analytic spaces
In this talk, I will explain some recent developments in the minimal model theory for projective morphisms between complex analytic spaces. Over the past few years, we have succeeded in establishing a framework for minimal model theory in this setting. I hope this framework will prove useful in the study of complex analytic singularities and the degenerations of projective varieties.
Andrea Fanelli. Maximal connected algebraic subgroups of the Cremona groups
One decade ago, Jérémy Blanc and Jean-Philippe Furter proved that the Cremona group of rank $n\ge2$ cannot be endowed with the structure of an (ind-)algebraic group. Since then, several people developed new approaches to study maximal connected algebraic subgroups of the Cremona groups, via modern birational geometry: those subgroups measure in some sense the complexity of Cremona groups. In this talk I will survey some progress in this field.
Jack Hall. Finiteness of integral transforms
If X is a smooth projective variety over a field k, then Beilinson (1978) showed that the bounded derived category of X could be identified as the subcategory of the unbounded derived category of quasi-coherent sheaves on X satisfying natural finiteness conditions. I will discuss some generalizations of this result to certain types of algebraic stacks.
David Swinarski. Invariant polynomials and singular curves in Mukai's models
In the 1990s Mukai introduced birational models of the moduli spaces of curves, one in each genus 7, 8, and 9. We discuss techniques for producing invariant polynomials related to these models and establish that several singular curves are GIT semistable in these models.
Piotr Pokora. Free curves in algebraic geometry
The main aim of the talk is to present the role of free curves in algebraic geometry, mostly focusing the questions regarding constructions of algebraic surfaces having extreme properties, like having large Picard numbers, or very high total Tjurina numbers.
Dmitry Zakharov. Resolving the Prym-Torelli map in g=4 via the trigonal construction.
The Torelli map t_g:M_g->A_g associates to a smooth algebraic curve its Jacobian variety. By a result of Mumford and Namikawa, the Torelli map extends to a morphism from the Deligne--Mumford compactification of M_g to the second Voronoi compactification of A_g. The Prym--Torelli map p_g:R_g->A_{g-1} associates to an etale double cover of algebraic curves its Prym variety. Unlike the case of Jacobians, p_g does not extend to the natural compactification of R_g by admissible covers. The indeterminancy locus of p_g was completely characterized by Friedman and Smith. A natural question is to determine the blowups that are needed to extend p_g to the various toroidal compactifications of A_{g-1}, for example to the second Voronoi. I will explain how resolve the indeterminancy loci of the Prym--Torelli map p_g to the second Voronoi compactification in the case g=4. The main tool is the trigonal construction, which has algebraic and tropical versions. As a companion result, I will explain how to compute the second moment of the tropical Prym variety.
Anoop Singh. Line bundles on the moduli space of Lie algebroid connections over a curve
We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid (L) connections on a compact Riemann surface X of genus g ≥ 3. A smooth compactification of the moduli space of L-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants.
Yongnam Lee. Morphisms from a very general hypersurface
In this talk, we will talk about a non-binational surjective morphism from a very general hypersurface X to a normal projective variety Y. We first show Y is a Fano variety if the degree of the morphism is bigger than a constant C where C depends on the dimension and degree of X. Next we prove an optimal upper bound of the morphism which is degree of X provided that Y is factorial, degree of the morphism is prime and bigger than a constant E where E depends only on the dimension of X. Also we will show that Y is a projective space under some conditions. This is a joint work with Yujie Luo and De-Qi Zhang.
Taro Sano. On Hodge structures of compact complex manifolds with semistable degenerations
Compact Kähler manifolds satisfy several nice cohomological properties such as Hodge symmetry and Hodge-Riemann bilinear relations. Friedman and Li recently showed that non-Kähler Calabi-Yau 3-folds which are obtained by conifold transitions of projective ones satisfy such properties. I will present examples of non-Kähler Calabi-Yau manifolds with such properties by smoothing normal crossing varieties.
Stephen Coughlan. Threefolds near the Noether line and their moduli spaces
We characterize those three-dimensional algebraic varieties of general type which satisfy equality in the 3-fold Noether inequality. These varieties admit fibrations in so-called (1,2)-surfaces. We can use this structure to describe their moduli spaces, which have surprising properties, especially in comparison with the classical work of Horikawa on surfaces. Joint work with Y. Hu, R. Pignatelli, and T. Zhang.
Uriya First. Versality of Torsors: Beyond Fields
Let G be an algebraic group over a field k. A G-torsor E over a k-variety X is called versal if every G-torsor over a k-field is a specialization of E and there are many such specializations in a well-defined sense. The minimal possible dimension of a variety supporting a versal G-torsor is one way to define the essential dimension of G. I will discuss results which show that versality properties can go beyond torsors over fields. Specifically, for every natural number d, there are G-torsors over k-varieties that specialize to any G-torsor over a d-dimensional affine noetherian k-scheme. However, there is a limit to how far one can push this --- there is G-torsor which is versal for all affine k-schemes if and only if G is unipotent. Moreover, in the non-unipotent case, there are explicit lower bounds on the dimension of a G-torsor that is versal for all d-dimensional varieties. This opens the door for defining ``higher'' essential dimension. To demonstrate the dichotomy between the unipotent and non- unipotent case, consider G=Z/pZ. Then G-torsors are just cyclic degree-p Galois extensions. In characteristic p, it is well known that every such (commutative) ring extension is an Artin--Schreier extension, and thus a specialization of the generic Artin--Schreier extension; this is a manifestation of the fact that G is unipotent and there exists a G-torsor which is versal for all affine k-schemes. In characteristic not p, the group G is not unipotent, so our results say that for every fixed d, there is a cyclic degree-p Galois extensions of k-rings which specializes to all degree-p Galois extensions of d-dimensional k-rings, but no Galois extension of rings will have this property for all d. (Partially based on joint work with Mathieu Florence and Zev Rosengarten.)
Jianqi Liu. Applications of the factorization theorem of conformal blocks to vertex operator algebras
Luca Tasin. Sasaki-Einstein metrics on spheres
It is a classical problem in geometry to construct interesting metrics on spheres. Sasaki-Einstein metrics are the analogous of Kähler-Einstein metrics for odd dimensional real manifolds. I will report on a joint work with Yuchen Liu and Taro Sano in which we construct infinitely many Sasaki-Einstein metrics on odd-dimensional spheres that bound parallelizable manifolds, proving in this way conjectures of Boyer-Galicki-Kollár and Collins-Székelyhidi. The construction is based on showing the K-stability of certain Fano weighted orbifold hypersurfaces.
Ishan Banerjee. Monodromy of curves in an algebraic surface
This talk will be about the monodromy group associated to a family of algebraic curves in an algebraic surface as a subgroup of the mapping class group. I will start by surveying some older results in this area about the image of monodromy in the symplectic group. I will then discuss joint work with Nick Salter, where we describe the precise image of monodromy in the mapping class group in the special case of complete intersections.
Valentina Beorchia. Identification of quasi-homogeneous hypersurface singularities
We illustrate an unexpected general relation between the Jacobian syzygies of a projective hypersurface with only isolated singularities and the nature of its singularities. This allows to establish a new method for the identification of quasi-homogeneous hypersurface isolated singularities. The result gives an insight on how the geometry of the polar map is reflected in the Jacobian syzygies.
Tomasz Szemberg. Sets of points in projective spaces and their projections
The purpose of this talk is to introduce the concepts of geproci and geprofi subsets in projective spaces. A set of points in a projective space is said to have the geproci property if its projection from a general points in the ambient space on a hyperplane is a complete intersection. We will discuss the structure and construction methods for such sets in the projective space of dimension three. In higher dimensional projective spaces no non-trivial sets of points have been discovered. However, a related concept of geprofi property leads to a rich and unexpected geometry considerations. A set of points in a projective space has the geprofi property if its projection from a general point is a full intersection, i.e., an intersection of two subvarieties of complimentary dimensions. We will show some basic phenomena occurring for such sets already in the projective space of dimension four. This talk is based on recent joint works with Luca Chiantini, Lucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore and Justyna Szpond.
Taro Fujisawa. A new approach to the nilpotent orbit theorem via the L^2-extension theorem of Ohsawa--Takegosi type
I would like to present a new proof of a part of the nilpotent orbit theorem. In our proof, the L^2 extension theorem of Ohsawa-Takegoshi type plays an essential role. Also, I would like to discuss about further developments in this direction.
Luca Schaffler. Fineness and smoothness of a KSBA moduli of marked cubic surfaces
The moduli space of cubic surfaces marked by their $27$ lines admits multiple compactifications arising from different perspectives. By work of Gallardo-Kerr-Schaffler, it is known that Naruki’s cross-ratio compactification is isomorphic to the normalization of the Kollár, Shepherd-Barron, Alexeev (KSBA) compactification parametrizing pairs $\left(S,\left(\frac{1}{9}+\epsilon\right)D\right)$, where $D$ is the sum of the $27$ marked lines on $S$, along with their stable degenerations. In this talk, we show that the normalization assumption is unnecessary by proving that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is achieved by studying the automorphisms and the $\mathbb{Q}$-Gorenstein obstructions of the stable pairs it parametrizes. This is joint work with Hanlong Fang and Xian Wu.
Changho Han. Compact moduli of K3 surfaces via trigonal maps
K3 surfaces, as a generalization of elliptic curves, have a rich amount of geometric properties. For instance, there are K3 surfaces that are cyclic triple covers of rational surfaces; Artebani and Sarti classified such generic K3 surfaces depending on lattice invariants. Such K3 surfaces admit Kulikov and KSBA degenerations, each leading to toroidal and KSBA compactifications of the moduli spaces of such K3 surfaces. As joint works with Valery Alexeev, Anand Deopurkar, and Philip Engel, I will explain how to use trigonal curves (triple covers of rational curves) to obtain aforementioned degenerations, leading to more explicit understandings of boundaries of those compactifications: such as classifications of generic members and the dimensions.
Jayan Mukherjee. Extendability of general K3 surfaces without Gaussian maps and classification of non-prime Fano threefolds
In this talk, we introduce an approach to the question of extendability of projective varieties via degeneration to ribbons. As an application of these methods to give a new proof of optimal results on the extendability of general non-prime K3 surfaces, classification of non-prime Fano threefolds and Mukai varieties and the irreducibility of their Hilbert schemes. The methods in this article also show the non-extendability of prime K3 surfaces for infinitely many values of g, for example when g is of the form g = 4k+1, k \geq 5. This involves degenerations of K3 surfaces to ribbons on embedded Hirzebruch surfaces, called K3 carpets. We directly give optimal upper bounds on the cohomology of the twisted normal bundle of the K3 carpets instead of computing coranks of Gaussian maps of the canonical curve sections. As a result of independent interest, we show such K3 carpets also appear as degenerations of smoothable simple normal crossings of two Hirzebruch surfaces embedded by arbitrary linear series intersecting along an anticanonical elliptic curve. Such type II degenerations constitute a smooth locus of codimension 6 in the Hilbert scheme of K3 surfaces.
Locations
Contributed talk sessions will happen in the following classrooms. Here is a link to the CSU campus map, and you may click on each classroom to see its location.