Titles and abstracts of the short talks

(in alphabetical order with respect to the family name of the speaker)

Toroidalization of locally toroidal morphisms (Razieh Ahmadian) (cancelled)

Toroidal varieties are algebraic varieties that are locally (formally) toric in structure, and toroidal morphisms are those morphisms of varieties which are locally determined by toric morphisms. The problem of toroidalization, proposed first in [1], is to construct a toroidal lifting of a dominant morphism φ:X→Y of algebraic varieties by blowing up nonsingular subvarieties in the target and domain. This problem is evidently very difficult, and it has been solved only when Y is a curve, or when φ is dominant and X, Y are of dimension ≤3 – see [3]. We provide a comprehensive survey of the toroidalization problem. In addition, we present some recent results in toroidalization of locally toroidal morphisms [2], which is among patching type problems.

References

  • 1. Abramovich, D., Karu, K., Matsuki, K. and Wlodarczyk, J., Torification and factorization of birational maps, JAMS 15 (2002), 531 – 572.
  • 2. Ahmadian, R., Toroidalization of locally toroidal morphisms,Trans. Amer. Math. Soc., Vol 375, no. 4 (2022), 2949 – 2986.
  • 3. Cutkosky, S.D., Toroidalization of dominant morphisms of 3-folds, Mem. Amer. Math. Soc., vol. 190, no. 890 (2007).

Toric reduction of singularities for Newton non-degenerate p-forms (Bilal Balo)

We prove the existence of reduction of singularities of p-forms satisfying a non-degeneracy condition expressed through their Newton polyhedron.

The Zariski-Lipman conjecture (Paul Barajas)

The Zariski-Lipman conjecture states that a complex variety with a locally free tangent sheaf is necessarily smooth. This conjecture has been proven in particular cases, such as hypersurfaces, homogeneous complete intersections, local complete intersections, log canonical spaces, and others. The objective of this talk is to review various known results concerning the Zariski-Lipman conjecture. We will then analyze this problem from a geometric perspective and present some new results in the case of normal surface singularities.

Bernstein-Sato polynomials of hyperplane arrangements in three variables (Daniel Bath)

Bernstein—Sato polynomials are famous invariants of hypersurfaces singularities, but famously difficult to compute. Hyperplane arrangements are, at least naively, easy to understand. One can hope naive simplicity outweighs famous difficulty; that is, one can hope Bernstein—Sato polynomials of hyperplane arrangements are easy to grok. In 2015, Walther showed this is not so easy: he revealed two arrangements in C3 with the same combinatorics but with different Bernstein—Sato polynomials root sets. We give a formula for the roots of the Bernstein—Sato polynomial for any arrangement in C3 . It turns out all but one candidate root are (easily) combinatorially determined. And the pathological candidate’s root-hood is equivalent to a simple algebraic criterion involving local cohomology of the Milnor algebra. This is the main prize in a larger study of Bernstein—Sato polynomials of locally everywhere quasi-homogeneous polynomials in C3.

This talk is based on:
Daniel Bath. Bernstein–Sato polynomials of locally quasi-homogeneous divisors in C3. arXiv e-prints 2402.08342, February 2024. (to appear in Compositio Mathematica)

Seminormal toric varieties (François Bernard)

Toric varieties are traditionally assumed to be normal due to the equivalence of categories between normal toric varieties and fans, which are combinatorial objects. In this talk, I will present a joint work with Antoine Boivin, where we study seminormal toric varieties. The seminormalization of an algebraic variety is a variant of normalization that remains bijective with the original variety. Building on a theorem by Reid and Roberts, which shows, in the affine case, that seminormalization is achieved by saturating each face of the associated monoid, and on a construction by Teissier and González Pérez to deal with non-affine and non-normal toric varieties, we establish an equivalence of categories between seminormal toric varieties and some combinatorial objects that we call ”fans with attached groups.” This provides a class of toric varieties with more general singularities than the normal ones, while having a much simpler combinatorial structure compared to general non-normal, non-affine toric varieties.

Reference
F. Bernard, A. Boivin, Seminormal toric varieties, arXiv :2412.13789

Cohomologies of p-group covers (Jędrzej Garnek)

Studying cohomology of a variety with an action of a finite group is a classical and well-researched topic. However, most of the previous results focus either on the tame ramification case, on some special groups, or on specific curves. In the talk, we will consider the case of a curve over a field of characteristic p with an action of a finite p-group. Our research suggests that the Hodge and de Rham cohomologies decompose as sums of certain ’local’ and ’global’ parts. The global part should be determined by the ’topology’ of the cover, while the local parts should depend only on an analytical neighbourhood of the fixed points of the action. In fact, the local parts should come from cohomologies of Harbater-Katz-Gabber curves, i.e. covers of the projective line ramified only over ∞. The motivation for such a decomposition comes from degenerating the smooth cover to a cover of nodal curves. During the talk, we present our results towards the proof of this conjecture. We also show some applications. Finally, we address the case of extending the results to nodal curves.

Related publications:

Stratification by the poles of the complex zeta function of µ-constant plane branch deformations (Roger Gómez López)

The complex zeta function is an analytic family of distributions which has a meromorphic extension to C. Its poles are related with the roots of the Bernstein-Sato polynomial. We study the stratification by the poles of the complex zeta function of μ-constant deformations of a plane branch. The results we obtain also enable the explicit computation of stratifications without relying on Gröbner bases.

References

  • [BGL] G. Blanco and R. Gómez-López, Stratification of μ-constant plane branch deformations by the poles of the complex zeta function, Work in progress.
  • [Bla19] G. Blanco, Poles of the complex zeta function of a plane curve, Adv. Math. 350 (2019), 396–439. MR 3947649
  • [CN91] P. Cassou-Noguès, Courbes de semi-groupe donné, Rev. Mat. Univ. Complut. Madrid 4 (1991), no. 1, 13–44. MR 1142547

On wheel relations for the K-theoretic Hall algebra of commuting variety. (Danil Gubarevich)

Consider the set of pairs of commuting complex n×n matrices (the commuting variety). It is preserved under the conjugation action of GLn(C) and has a fixed point (0,0). The direct sum (over n) of the equivariant K-theories of these commuting varieties is equipped with an associative multiplication and admits an embedding into the K-theory of a point (i.e., into the algebra of symmetric polynomials), via restriction to the fixed point.
It was observed by Y. Zhao (arXiv:1909.07870 ) that the image of this embedding is described by the so-called wheel relations. 
In this talk, I will show how to obtain similar relations for a general reductive group.

Variation of the local fundamental group on a complex normal space (Sudarshan Gurjar)

For a complex normal space and a point on it, there is a notion of the local fundamental group at that point. We discuss the variation of these local fundamental groups at points on a complex, normal space. This variation is not semicontinuous in general. In this direction, we show that finite Galois descent of upper semicontinuity of the local fundamental group holds at a factorial complex analytic germ. We also show by an example that the corresponding result for the local first homology group is not true in general. As an application of the upper semicontinuity of local fundamental groups, we will show that the singular locus of a Gorenstein complex analytic space with rational singularities has singular locus in codimension 3 at a point with trivial local fundamental group. We also show that the local fundamental groups that occur at points on a normal complex algebraic variety are finite in number upto isomorphism. This is based on joint work with R.V Gurjar and Buddhadev Hajra.

A new proof of monomialisation from 3-folds to surfaces (Yueting Jiang)

In this paper, we give a new proof of the foundational result, due to S. Cutkosky, on the existence of a monomialisation of a morphism from a 3-fold to a surface. Our proof brings to the fore the notion of log-Fitting ideals, and requires us to develop new methods related to Rank Theorems and log-Fitting ideals.

Multiplicative Chow-Kunneth decomposition for nested Hilbert-Schemes (Inder Kaur)

The Chow ring of a variety encodes a lot of information about its geometry and is the subject of many interesting conjectures. A conjecture of Shen-Vial predicts that the Chow ring of any hyperkaehler variety admits a multiplicative Chow-Kunneth (MCK) decomposition. Shen and Vial show that if a variety X admits MCK decompoition and so does a smooth subvariety Y⊂X, then the blow-up of X along Y also admits MCK decomposition. In this talk I will discuss whether the blow-up of X along a singular subvariety Y , admits a MCK decomposition and apply this to the case of nested Hilbert schemes of points on a K3/abelian surface. This is joint work with R. Laterveer.

Hodge modules on toric varieties (Hyunsuk Kim)

We study the structure of some natural Hodge modules on toric varieties, such as the intersection cohomology Hodge module and the trivial Hodge module. By giving a precise description of these objects as Hodge modules, we obtain new results concerning local cohomology modules, local cohomological dimension, depth of reflexive differentials, and the Hodge structure of singular cohomologies for toric varieties, which are related to delicate Hodge theoretic invariants of singularities. We also give some applications to combinatorics of convex polyhedral cones. This is a joint work with Sridhar Venkatesh. The first part of the work is on [KV24] and there will be an upcoming work [KV] soon.

References

  • [KV] Hyunsuk Kim and Sridhar Venkatesh, The local cohomology of toric singularities (in preparation).
  • [KV4] Hyunsuk Kim and Sridhar Venkatesh, The intersection cohomology Hodge module of toric varieties, arXiv:2404.04767 (2024).

Bergman spaces on algebraic curves (Alexander A. Kubasch)

Bergman spaces emerge from functional analysis and operator theory in several complex variables and arise in physics as quantum Hilbert spaces. They have been studied extensively over the past half century, but have never been considered on singular spaces. The Bergman space A2(U) of an open set U⊆Cn is the Hilbert space of square-integrable holomorphic functions on U. If n≥2, then dim A2(U) can attain any non-negative integer value as well as infinity. In dimension n=1 however, the Bergman space of any open set U⊆C is either infinite-dimensional or trivial [W84].
In this talk I will explore Bergman spaces on quasi-projective algebraic curves. It turns out that the dichotomy dim A2∈{0,∞} only holds for an affine curve if the complexity of its singularities can be controlled by its behavior at infinity. This is the consequence of a broader phenomenon that holds for all quasi-projective curves generalizing results of [GGV22, GGV24, S22]. I will also introduce an “L2-version” of the δ-invariant and compare it to the usual one. 
Joint work with L. Koltai and R. Szőke. 

References

  • [GGV22] A.-K. Gallagher, P. Gupta, L. Vivas: On the dimension of Bergman spaces on P1 , La Matematica, 1(3), 666-684, 2022
  • [GGV24] A.-K. Gallagher, P. Gupta, L. Vivas: On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces, Indiana U. Math J, Vol 73, (4), 1541-1549, 2024
  • [S22] R. Szőke: On a Theorem of Wiegerinck Anal. Math. 48, 581–587 2022
  • [KKS25] L. Koltai, A. A. Kubasch, R. Szőke: Bergman spaces on algebraic curves, arXiv:2504.02341
  • [W84] J.J.O.O. Wiegerinck: Domains with finite dimensional Bergman space, Math. Z. 559-562, 1984

On the Nash Problem over 3-fold Terminal Singularities (Keng-Hung Steven Lin)

Let X be a singular algebraic variety over the complex numbers. Nash proposed to use irreducible families of arcs based on the singular locus of X [Nas95], inducing the so-called Nash valuations, to characterize divisorial valuations appearing in every resolution of singularities over X, which are called essential valuations. In particular, Nash proves that every Nash valuation is essential, and it is natural to ask if every essential valuation corresponds to some Nash valuation. This is known as the Nash problem for X. The Nash problem is known to be true in dimensions 1 and 2 [FdBPP12], in several interesting cases in arbitrary dimensions [IK03, JK13], but false in general starting in dimensions 3 [dF13]. Apart from the Nash problem, it is an interesting problem to characterize Nash valuations over a singular variety [dFD16, Che23]. In this talk, I will briefly introduce the Nash problem, discuss the most relevant developments, and then talk about my current work (in progress) on the Nash problem over 3-fold terminal singularities.

References

  • [Che16] Jungkai Alfred Chen. Explicit resolution of three dimensional terminal singularities. In Minimal models and extremal rays (Kyoto, 2011), volume 70 of Adv. Stud. Pure Math., pages 323–360. Math. Soc. Japan, [Tokyo], 2016.
  • [Che23] Hsin-Ku Chen. On the Nash problem for terminal threefolds of type cA/r. Internat. J. Math., 34(10):Paper No. 2350055, 29, 2023.
  • [dF13] Tommaso de Fernex. Three-dimensional counter-examples to the Nash problem. Compos. Math., 149(9):1519–1534, 2013.
  • [dFD16] Tommaso de Fernex and Roi Docampo. Terminal valuations and the Nash problem. Invent. Math., 203(1):303–331, 2016.
  • [FdBPP12] Javier Fernández de Bobadilla and Marı́a Pe Pereira. The Nash problem for surfaces. Ann. of Math. (2), 176(3):2003–2029, 2012.
  • [Hay99] Takayuki Hayakawa. Blowing ups of 3-dimensional terminal singularities. Publ. Res. Inst. Math. Sci., 35(3):515–570, 1999.
  • [Hay00] Takayuki Hayakawa. Blowing ups of 3-dimensional terminal singularities. II. Publ. Res. Inst. Math. Sci., 36(3):423–456, 2000.
  • [IK03] Shihoko Ishii and János Kollár. The Nash problem on arc families of singularities. Duke Math. J.,120(3):601–620, 2003.
  • [JK13] Jennifer M. Johnson and János Kollár. Arc spaces of cA-type singularities. J. Singul., 7:238–252, 2013.
  • [Mor85] Shigefumi Mori. On 3-dimensional terminal singularities. Nagoya Math. J., 98:43–66, 1985.
  • [Nas95] John F. Nash, Jr. Arc structure of singularities. Volume 81, pages 31–38. 1995. A celebration of John F. Nash, Jr.

Classification of T-singular surfaces with small K2/Pg (Vincente Monreal)

In this talk I will discuss recent work on the classification of complex surfaces with only T-singularities, ample dualizing sheaf, and K2≤2Pg−4. This work provides the first classification of singular surfaces allowing arbitrary values of pg and an unrestricted number of singularities; It also implies that the challenging Horikawa problem cannot be addressed through complex T-degenerations, and proposes new questions regarding diffeomorphism types. This is joint work with Giancarlo Urzúa and Jaime Negrete.
Content:
• When Can We Classify? Introduction to Geoegraphy of Surfaces: Overview of Gieseker moduli spaces of surfaces of general type [Gie77] and their KSBA compactifications [Ale94, KSB88].
• Why Should We Classify? Introduction to the Horikawa problem [Hor76] and an approach inspired by Manetti’s work [Man01].
• How Do We Classify? Summary of the main techniques used for classifying both smooth and singular complex surfaces with ample dualizing sheaf [Hor76, CFPR24, ESU24, RU19, FRU23]. Presentation of our methods [MNU24] and discussion of the results (properties of the surfaces, connection with our motivation, geography results).

  • [Ale94] V. Alexeev, Boundedness and K2 for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810.
  • [CFPR24] S. Coughlan, M. Franciosi, R. Pardini, and S. Rollenske, 2-Gorenstein stable surfaces with KX2 = 1 and χ(X) = 3, Preprint arXiv:2409.07854 (2024).
  • [ESU24] J. Evans, A. Simonetti, and G. Urzúa, Tropical methods for stable Horikawa surfaces, Preprint arXiv:2405.02735 (2024).
  • [RU19] J. Rana and G. Urzúa, Optimal bounds for T-singularities in stable surfaces, Adv. Math. 345 (2019),814–844.
  • [FRU23] F. Figueroa, J. Rana, and G. Urzua, Optimal bounds for many T-singularities in stable surfaces, Preprint arXiv:2308.05624 (2023).
  • [Gie77] D. Gieseker, Global moduli for surfaces of general type, Invent. Math. 43 (1977), no. 3, 233–282.
  • [Hor76] E. Horikawa, Algebraic surfaces of general type with small C12. Ann. of Math. (2) 104 (1976), no. 2, 357–387.
  • [KSB88] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338.
  • [Man01] M. Manetti, On the moduli space of diffeomorphic algebraic surfaces, Invent. Math. 143 (2001), no. 1, 29–76.
  • [MNU24] V. Monreal, J. Negrete, and G. Urzúa, Classification of Horikawa surfaces with T-singularities, arXiv preprint arXiv:2410.02943 (2024).

An algebraical geometrical and topological approach to 2-dimensional Jacobian Conjecture and a proof of the complex conjecture until degree 104 (Thuy Nguyen)

Introducing so-called “non-proper variables”, we classify the non-proper maps F : C2 → C2 into two classes: the first class satisfies the Jacobian conjecture, while if there exists a Keller map that is non-proper (counter-example for the 2-dimensional complex Jacobian conjecture), it belongs to the second class. Moreover, by Newton polygon approach, we prove the 2-dimensional complex Jacobian conjecture until degree 104, improving Moh boundary (1983) from 100 till 104. 

Reference
Thuy Nguyen, Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104. To appear in Quaestiones Mathematicae. ArXiv:1902.05923.

Counting plane curves with δ nodes and a triple point (Anantadulal Paul)

The enumeration of plane curves with δ nodes was an intriguing question in enumeration geometry. It is well understood from various perspectives. However, the enumeration of curves with degenerate singularities (more degenerate than nodes) is challenging. In this talk, we shall discuss how to enumerate the number Nd(A1δD4) defined as the number of plane degree d curves with δ many nodes and one triple point (satisfying the correct number of point insertions). The number Nd(A1δD4) is known when the total codimension is less or equal to 8, i.e., 0≤δ≤4. We present our work (in progress), indicating a solution for all δ.

Motivic local density in non-Archimedean geometry (Sidonie Ratajczak)

We consider a field K equipped with a distance and a measure. The local density of a set X in Kn at a point is defined as the limit, if it exists, of local volumes normalized. Intuitively, local density measures how the set fills the space locally. This notion is interesting when we look at a singular point of a variety. We can generalize this notion to valued fields for which there is no classical measure theory, as C((t)), by using motivic integration. The motivic volume is no longer a real number but an element of the Grothendieck group of varieties. The goal of this talk is to introduce the notion of local density at a singular point of a variety defined over C((t)), after giving an overview of motivic integration.

References

  • [1] Raf Cluckers, Georges Comte, François Loeser, Local metric properties and regular stratifications of p-adic definable sets, Comment. Math. Helv., 2009.
  • [2] Raf Cluckers, François Loeser, Constructible motivic functions and motivic integration, Annals of Math., 2010
  • [3] Richard Draper, Intersection theory in analytic geometry, Math. Ann., 1969.
  • [4] Arthur Forey, Motivic local density, Mathematische Zeitschrift, 2017.
  • [5] Krzysztof Kurdyka, Gilles Raby, Densité des ensembles sous-analytiques, Ann. Inst. Fourier, 1989.
  • [6] Pierre Lelong, Intégration sur un ensemble analytique complexe, Ann. Inst. Fourier, 1989.

Categorification with Lattice Homology (Gergő Schefler)

Lattice (co)homology is a connecting bridge between singularity theory and several other mathematical topics: its different variants make connections to low dimensional topology, analytic invariants and commutative algebra. Recently T. Ágoston and A. Némethi introduced some combinatorial requirements, satisfying which the Euler characteristic of the resulting lattice (co)homology can be prescribed [1]. They used this recipe to categorify the delta invariant of curve singularities [2] and the geometric genus of normal surface singularities [1, 3]. In this talk I will present a common generalization of their work, building on the valuative origins of their constructions. More specifically, I will define the (symmetric) lattice homology of finite codimensional integrally closed (ideals and) submodules (categorifying their codimension). I will discuss well-definedness and some immediate corollaries such as lower bounds, reduction-type theorems for surfaces singularities and relations to deformations. The results come from ongoing joint work with A. Némethi.

References

  • [1] Ágoston, T., and Némethi, A.: Analytic lattice cohomology of surface singularities, arXiv:2108.12294 (2021).
  • [2] Ágoston, T., and Némethi, A.: The analytic lattice cohomology of isolated curve singularities, arXiv:2301.08981 (2023).
  • [3] Némethi, A.: Normal Surface Singularities, Springer Nature, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol 74 (2022).