Calf Seminar
About the Calf
Calf is the junior COW, an algebraic geometry seminar group primarily aimed at PhD students.
The organisers for academic year 2024/25 are Heath Pearson (Nottingham), Siao Chi Mok (Cambridge), and Alexander Fruh (Birmingham), as well as an extensive network of local organisers at different universities. Calf announcements are made using the COW mailing list. If you would like to get involved in the organisation, or suggest your institution as the next venue, please contact any of the people named above.
The COW seminar has some funding for travel expenses, and information on reimbursement can be found on the main COW webpage.
About the Calf
Calf is the junior COW, an algebraic geometry seminar group primarily aimed at PhD students.
The organisers for academic year 2024/25 are Heath Pearson (Nottingham), Siao Chi Mok (Cambridge), and Alexander Fruh (Birmingham), as well as an extensive network of local organisers at different universities. Calf announcements are made using the COW mailing list. If you would like to get involved in the organisation, or suggest your institution as the next venue, please contact any of the people named above.
The COW seminar has some funding for travel expenses, and information on reimbursement can be found on the main COW webpage.
Upcoming Meeting:
University of Cambridge, Friday 17th January 2025
MR4, Centre for Mathematical Sciences, 13:00 - 17:00
13:00: Terry Song (University of Cambridge)
Title: Genus one stable maps and Vakil—Zinger moduli space
Abstract: In this talk I will introduce the Kontsevich space of genus one stable maps and desingularisation of its main component, which is constructed by Vakil—Zinger and admits an interpretation in logarithmic geometry by Ranganathan—Santos-Parker—Wise. If time permits, I will present calculations on Euler characteristics and Betti numbers that offer a quantitative comparison between the two moduli spaces. Partly based on joint work with Siddarth Kannan.
14:30: Patience Ablett (University of Warwick)
Title: Gotzmann's persistence theorem for smooth projective toric varieties
Abstract: Gotzmann's persistence theorem is a useful tool for finding equations of the Hilbert scheme parameterising subschemes of projective space. From the commutative algebra perspective, there is a natural way to generalise such Hilbert schemes to any smooth projective toric variety. A key example we will discuss is the Hilbert scheme parameterising subschemes of the product of projective spaces. We will see how Gotzmann's persistence theorem generalises to this setting.
16:00: Thamarai Valli Venkatachalam (UCL)
Title: Complete intersections in toric varieties
Abstract: Toric varieties, with their rich combinatorial structure, simplify computations, and their complete intersections inherit these advantages. In this talk, I will introduce toric varieties as GIT quotients and explore complete intersections within these varieties, along with their combinatorial properties. We will also examine how these properties aid in classifying Fano varieties. If time permits, I will discuss my ongoing work on the classification of Fano complete intersections in toric varieties.
University of Cambridge, Friday 17th January 2025
MR4, Centre for Mathematical Sciences, 13:00 - 17:00
13:00: Terry Song (University of Cambridge)
Title: Genus one stable maps and Vakil—Zinger moduli space
Abstract: In this talk I will introduce the Kontsevich space of genus one stable maps and desingularisation of its main component, which is constructed by Vakil—Zinger and admits an interpretation in logarithmic geometry by Ranganathan—Santos-Parker—Wise. If time permits, I will present calculations on Euler characteristics and Betti numbers that offer a quantitative comparison between the two moduli spaces. Partly based on joint work with Siddarth Kannan.
14:30: Patience Ablett (University of Warwick)
Title: Gotzmann's persistence theorem for smooth projective toric varieties
Abstract: Gotzmann's persistence theorem is a useful tool for finding equations of the Hilbert scheme parameterising subschemes of projective space. From the commutative algebra perspective, there is a natural way to generalise such Hilbert schemes to any smooth projective toric variety. A key example we will discuss is the Hilbert scheme parameterising subschemes of the product of projective spaces. We will see how Gotzmann's persistence theorem generalises to this setting.
16:00: Thamarai Valli Venkatachalam (UCL)
Title: Complete intersections in toric varieties
Abstract: Toric varieties, with their rich combinatorial structure, simplify computations, and their complete intersections inherit these advantages. In this talk, I will introduce toric varieties as GIT quotients and explore complete intersections within these varieties, along with their combinatorial properties. We will also examine how these properties aid in classifying Fano varieties. If time permits, I will discuss my ongoing work on the classification of Fano complete intersections in toric varieties.
Recent Meeting:
University of Warwick, Friday 29th November 2024
Room B1.01, Zeeman Building, 14:00 - 18:00
Local Organiser: Marc Truter
14:00: Menelaos Zikidis (University of Sheffield)
Title: Joyce Structures on Spaces of Bridgeland Stability Conditions
Abstract: In this lecture I will try to convey the essence behind the notion of a Joyce structure, concentrating on fundamental examples. Joyce structures are geometric structures on spaces of Bridgeland stability conditions Stab(D) aiming to encode geometrically Donaldson-Thomas invariants of triangulated Calabi-Yau 3 categories D. The theory can be made explicit in a large class of examples where Stab(D) is identified with the moduli of meromorphic quadratic differentials on algebraic curves. In this case the geometric structure arises, via isomonodromic deformations of Stokes-Riemann-Hilbert data, as a family of non-linear Ehresmann connections on a modulispace parametrizing wild algebraic curves, wild parabolic bundles, irregular Higgs fields and flat connections. Such a Joyce structure equips this space of irregular objects with a Complex Hyperkähler structure and an associated Twistor space.
15:30: James Jones (Loughborough University)
Title: Type II Degenerations of Degree 4 K3 Surfaces
Abstract: Moduli spaces of K3 surfaces have long been studied, beginning with the proof of the Torelli theorem for K3s in the 70s. The most well-known compactification of this moduli space is still probably the Baily-Borel compactification, whose boundary components consist of Type II and Type III degenerations. Meanwhile, the GIT compactification is able to explicitly describe its boundary components, often by equations as studied by Shah in the 80s and more recently by Laza and O'Grady. In this talk we provide the requisite definitions of degenerations, giving examples along the way. The punchline of the talk is an explicit classification of all Type II degenerations of degree 4 K3 surfaces.
17:00: Aporva Varshney (University College London)
Title: The Derived Category of a Singular Variety
Abstract: When varieties become singular, they wreak havoc on the derived category. Nonetheless, we must reckon with singularities in some way: since we started caring about things like minimal models and moduli spaces, they’ve become all too prevalent, and derived categories can be a very useful tool to study geometry. I’ll give a (hopefully) accessible introduction on how we can reconcile these two foes: first we’ll look at an example of a nodal curve and explore what goes wrong in the derived category. This will lead us to the idea of “categorical absorption” (following work of Kuznetsov-Shinder) which breaks down the derived category into a smooth component and a singular component. Importantly for nodal varieties, this singular component behaves nicely under smoothings. Time permitting, we’ll also look at a small generalization to non-isolated singularities.
University of Warwick, Friday 29th November 2024
Room B1.01, Zeeman Building, 14:00 - 18:00
Local Organiser: Marc Truter
14:00: Menelaos Zikidis (University of Sheffield)
Title: Joyce Structures on Spaces of Bridgeland Stability Conditions
Abstract: In this lecture I will try to convey the essence behind the notion of a Joyce structure, concentrating on fundamental examples. Joyce structures are geometric structures on spaces of Bridgeland stability conditions Stab(D) aiming to encode geometrically Donaldson-Thomas invariants of triangulated Calabi-Yau 3 categories D. The theory can be made explicit in a large class of examples where Stab(D) is identified with the moduli of meromorphic quadratic differentials on algebraic curves. In this case the geometric structure arises, via isomonodromic deformations of Stokes-Riemann-Hilbert data, as a family of non-linear Ehresmann connections on a modulispace parametrizing wild algebraic curves, wild parabolic bundles, irregular Higgs fields and flat connections. Such a Joyce structure equips this space of irregular objects with a Complex Hyperkähler structure and an associated Twistor space.
15:30: James Jones (Loughborough University)
Title: Type II Degenerations of Degree 4 K3 Surfaces
Abstract: Moduli spaces of K3 surfaces have long been studied, beginning with the proof of the Torelli theorem for K3s in the 70s. The most well-known compactification of this moduli space is still probably the Baily-Borel compactification, whose boundary components consist of Type II and Type III degenerations. Meanwhile, the GIT compactification is able to explicitly describe its boundary components, often by equations as studied by Shah in the 80s and more recently by Laza and O'Grady. In this talk we provide the requisite definitions of degenerations, giving examples along the way. The punchline of the talk is an explicit classification of all Type II degenerations of degree 4 K3 surfaces.
17:00: Aporva Varshney (University College London)
Title: The Derived Category of a Singular Variety
Abstract: When varieties become singular, they wreak havoc on the derived category. Nonetheless, we must reckon with singularities in some way: since we started caring about things like minimal models and moduli spaces, they’ve become all too prevalent, and derived categories can be a very useful tool to study geometry. I’ll give a (hopefully) accessible introduction on how we can reconcile these two foes: first we’ll look at an example of a nodal curve and explore what goes wrong in the derived category. This will lead us to the idea of “categorical absorption” (following work of Kuznetsov-Shinder) which breaks down the derived category into a smooth component and a singular component. Importantly for nodal varieties, this singular component behaves nicely under smoothings. Time permitting, we’ll also look at a small generalization to non-isolated singularities.
