Organizing Committee
- Melody Chan
Brown University - Antonella Grassi
University of Pennsylvania and Università di Bologna - Rohini Ramadas
Brown University - Julie Rana
Lawrence University - Isabel Vogt
Stanford University
Abstract
The Women in Algebraic Geometry Collaborative Research Workshop will bring together researchers in algebraic geometry to work in groups of 4-6, each led by one or two senior mathematicians. The goals of this workshop are: to advance the frontiers of modern algebraic geometry, including through explicit computations and experimentation, and to strengthen the community of women and non-binary mathematicians working in algebraic geometry. This workshop capitalizes on momentum from a series of recent events for women in algebraic geometry, starting in 2015 with the IAS Program for Women in Mathematics on algebraic geometry.
Successful applicants will be assigned to a group based on their research interests. The groups will work on open-ended projects in diverse areas of current interest, including moduli spaces and combinatorics, degenerations, and birational geometry. Several of the proposed projects extensively involve experimentation and computation, which will increase the likelihood that concrete progress is made over the course of five days and provide useful training in computational mathematics.
In their personal statements, applicants should rank in order their top three choices of projects. They should also address their familiarity with the suggested prerequisites.
Application Deadline: February 9, 2020
![Image for "VIRTUAL ONLY: Women in Algebraic Geometry"](http://app.icerm.brown.edu/img/299_image.jpg)
Confirmed Speakers & Participants
Talks will be presented virtually or in-person as indicated in the schedule below.
- Speaker
- Poster Presenter
- Attendee
- Virtual Attendee
-
Alejandra Rincón-Hidalgo
ICTP- International center for theoretical physics.
-
Carolina Araujo
IMPA
-
Roya Beheshti Zavareh
Washington University
-
Renee Bell
Université Paris-Sud
-
Emma Brakkee
University of Amsterdam
-
Madeline Brandt
Brown University
-
Juliette Bruce
University of Wisconsin - Madison
-
Chiara Camere
University of Milano
-
Ana-Maria Castravet
University of Versailles, France
-
Melody Chan
Brown University
-
Linda Chen
Swarthmore College
-
Man-Wai Cheung
Harvard University
-
Emily Clader
San Francisco State University
-
Paola Comparin
UFRO Universidad de La Frontera
-
María Angélica Cueto
Ohio State University
-
Chiara Damiolini
Princeton University
-
Kristin DeVleming
University of California, San Diego
-
Sarah Frei
Rice University
-
Angela Gibney
Rutgers University, New Brunswick
-
Antonella Grassi
University of Pennsylvania and Università di Bologna
-
Annalisa Grossi
Augsburg University
-
Lauren Heller
University of California, Berkeley
-
Daoji Huang
Cornell University
-
Kelly Jabbusch
University of Michigan - Dearborn
-
Lena Ji
Princeton University
-
Elana Kalashnikov
Harvard University
-
Amanda Knecht
Villanova University
-
Hannah Larson
Stanford University
-
Shiyue Li
Brown University
-
Jennifer Li
UMass Amherst
-
Svetlana Makarova
MIT
-
Enrica Mazzon
Max Planck Institute for Mathematics
-
Margarida Melo
Universita Roma Tre
-
Kalina Mincheva
Yale University
-
Laura Pertusi
Università degli Studi di Milano
-
Rohini Ramadas
Brown University
-
Julie Rana
Lawrence University
-
Giulia Saccà
Columbia University
-
Cecília Salgado
UFRJ
-
Soumya Sankar
University of Wisconsin-Madison
-
Alessandra Sarti
Université de Poitiers
-
Aleksandra Sobieska
Texas A&M University
-
Libby Taylor
Stanford University
-
Alexandra Viktorova
Stony Brook University
-
Bianca Viray
University of Washington
-
Nivedita Viswanathan
The University of Edinburgh
-
Isabel Vogt
Stanford University
-
Rachel Webb
University of Michigan
-
Corey Wolfe
Tulane University
-
Weihong Xu
Rutgers
-
Lei Yang
Northeastern University
-
Claudia Yun
Brown University
-
Aline Zanardini
University of Pennsylvania
-
Yan Zhou
Beijing International Center for Mathematical Research, Peking University
Workshop Schedule
Monday, July 27, 2020
Time | Event | Location | Materials |
---|---|---|---|
10:50 - 11:00am EDT | Welcome - ICERM Director | ||
11:00 - 11:30am EDT | Organizer Welcome | ||
11:30 - 12:30pm EDT | Project Introductions ~5-Minutes/ Project - Project 1: Higher Fano manifolds Project 2: The rational cohomology of A_g Project 3: Moduli Spaces of Curves with Cyclic Action Project 4: Tropical degree-two del Pezzo surfaces and their 56 lines Project 5: Basepoint free cycles on the moduli space of genus zero curves Project 6: A hands-on introduction to aspects of the minimal model program and to Noether-Lefshetz problems Project 7: Hyper-Kähler manifolds, moduli spaces of sheaves, and Lagrangian fibrations Project 8: Automorphisms of K3 surfaces Project 9: Invariants of Fano threefolds over nonclosed fields | ||
12:30 - 2:00pm EDT | Unscheduled Project Time | ||
2:30 - 3:00pm EDT | "Make Your Own Coffee" Break |
Tuesday, July 28, 2020
Time | Event | Location | Materials |
---|---|---|---|
11:00 - 11:05am EDT | Math Introductions 5-minute talks - Emma Brakkee, University of Amsterdam | ||
11:05 - 11:10am EDT | Math Introductions 5-minute talks - Juliette Bruce, University of Wisconsin - Madison | ||
11:10 - 11:15am EDT | Math Introductions 5-minute talks - Daoji Huang, Cornell University | ||
11:15 - 11:20am EDT | Math Introductions 5-minute talks - Lena Ji, Princeton University | ||
11:20 - 11:25am EDT | Math Introductions 5-minute talks - Laura Pertusi, Università degli Studi di Milano | ||
11:25 - 11:30am EDT | Math Introductions 5-minute talks - Alejandra Rincón Hidalgo, ICTP- International center for theoretical physics | ||
11:30 - 12:00pm EDT | "Make Your Own Coffee" Break | ||
12:00 - 2:00pm EDT | Unscheduled Project Time | ||
3:00 - 3:30pm EDT | "Make Your Own Coffee" Break |
Wednesday, July 29, 2020
Time | Event | Location | Materials |
---|---|---|---|
11:00 - 11:30am EDT | "Make Your Own Coffee" Break | ||
12:00 - 2:00pm EDT | Unscheduled Project Time | ||
2:00 - 2:05pm EDT | Math Introductions 5-minute talks - Madeline Brandt, Brown University | ||
2:05 - 2:10pm EDT | Math Introductions 5-minute talks - Kristin DeVleming, University of California, San Diego | ||
2:10 - 2:15pm EDT | Math Introductions 5-minute talks - Sarah Frei, Rice University | ||
2:15 - 2:20pm EDT | Math Introductions 5-minute talks - Hannah Larson, Stanford University | ||
2:20 - 2:25pm EDT | Math Introductions 5-minute talks - Aleksandra Sobieska, Texas A&M University | ||
2:25 - 2:30pm EDT | Math Introductions 5-minute talks - Rachel Webb, University of Michigan | ||
2:30 - 3:00pm EDT | "Make Your Own Coffee" Break |
Thursday, July 30, 2020
Time | Event | Location | Materials |
---|---|---|---|
11:00 - 11:05am EDT | Math Introductions 5-minute talks - Shiyue Li, Brown University | ||
11:05 - 11:10am EDT | Math Introductions 5-minute talks - Jennifer Li, UMass Amherst | ||
11:10 - 11:15am EDT | Math Introductions 5-minute talks - Svetlana Makarova, MIT | ||
11:15 - 11:20am EDT | Math Introductions 5-minute talks - Soumya Sankar, University of Wisconsin-Madison | ||
11:20 - 11:25am EDT | Math Introductions 5-minute talks - Alexandra Viktorova, Stony Brook University | ||
11:25 - 11:30am EDT | Math Introductions 5-minute talks - Weihong Xu, Rutgers | ||
11:30 - 11:35am EDT | Math Introductions 5-minute talks - Aline Zanardini, University of Pennsylvania | ||
11:35 - 12:00pm EDT | "Make Your Own Coffee" Break | ||
12:00 - 2:00pm EDT | Unscheduled Project Time | ||
3:00 - 3:30pm EDT | "Make Your Own Coffee" Break |
Friday, July 31, 2020
Time | Event | Location | Materials |
---|---|---|---|
11:00 - 11:30am EDT | "Make Your Own Coffee" Break | ||
2:00 - 3:30pm EDT | Project Summaries - 10 minutes per project - Project 9: Invariants of Fano threefolds over nonclosed fields Project 8: Automorphisms of K3 surfaces Project 7: Hyper-Kähler manifolds, moduli spaces of sheaves, and Lagrangian fibrations Project 6: A hands-on introduction to aspects of the minimal model program and to Noether-Lefshetz problems Project 5: Basepoint free cycles on the moduli space of genus zero curves Project 4: Tropical degree-two del Pezzo surfaces and their 56 lines Project 3: Moduli Spaces of Curves with Cyclic Action Project 2: The rational cohomology of A_g Project 1: Higher Fano manifolds | ||
3:30 - 4:00pm EDT | "Make Your Own Coffee" Break |
Lecture Videos
Math Introductions
Project Descriptions
Project 1: Higher Fano manifolds
Leadership: Carolina Araujo, (IMPA), Roya Beheshti (Washington University) & Ana-Maria Castravet (University of Versailles, France)
The project will be centered around investigating the structure of higher Fano manifolds. Fano manifolds are manifolds whose first Chern class intersects all curves positively. This condition has far reaching implications; for example, the cone of effective 1-cycles is rational polyhedral, and one-dimensional families of such manifolds admit sections. De Jong and Starr introduced 2-Fano manifolds in [2], as manifolds with both first and second Chern character positive. Conjecturally, 2-Fano manifolds have rational polyhedral cones of effective 2-cycles, and two-dimensional families of 2-Fano manifolds admit sections (provided there is no Brauer-Manin obstruction).
Some effort has been put into understanding the structure of 2-Fano varieties [1,3,4]. Currently, all known examples have Picard number one, and can be described as zero loci of sections of certain vector bundles. Furthermore, the 2-Fano condition has been studied for smooth toric varieties in [5]. In this project, we study the following question: is every smooth toric 2-Fano isomorphic to a projective space? This has been checked using a computer up to dimension 6, but one project would be to try to prove this, or possibly explore it in dimensions higher than 6.
One can similarly define 3-Fano manifolds. There has been little investigation of their structure. One project would be to go through the classification of 2-Fano varieties of high index in [2] and decide which ones are 3-Fano. Similarly, we may investigate toric 3-Fanos: can the new positivity condition be interpreted combinatorially as in the case of 2-Fanos in [5] and does that help to prove that the only toric 3-Fano manifolds are projective spaces?
Preferred background: Knowledge of algebraic geometry at the level of Hartshorne. Some additional knowledge of toric geometry will be useful, not expected at the time of application.
References
A. J. de Jong and J. Starr, A note on Fano manifolds whose second Chern character is positive, preprint arXiv:math/0602644v1 [math.AG], (2006).
A. J. de Jong and J. Starr, Higher Fano manifolds and rational surfaces, Duke Math. J. Vol. 139 (2007), no. 1, 173–183.
C. Araujo, A.-M. Castravet, Classification of 2-Fano manifolds with high index, A celebration of algebraic geometry, 1–36, Clay Math. Proc. 18, Amer. Math. Soc., Providence, RI, 2013
C. Araujo, A.-M. Castravet, Polarized minimal families of rational curves and higher Fano manifolds, Amer. J. of Math. 134, no. 1 (2012), 87–107 E. E. Nobili, Birational geometry of toric varieties, PH. D.thesis, arXiv: 1204.3883
Project 2: The rational cohomology of A_g
Leadership: Melody Chan (Brown University) & Margarida Melo (Università Roma Tre)
This project studies the moduli spaces A_g of principally polarized abelian varieties of dimension g using techniques from [CGP18]. Just as with M_g the orbifold Euler characteristic and the stable cohomology are classically understood [Bor74, Har71], but the full cohomology ring H^*(A_g;Q) is a mystery even for small g: g ≤ 2 is classical, g = 3 is the work of Hain [Hai02], and g ≥ 4 is already unknown. Computations in the 4 ≤ g ≤ 8 range would already be interesting and new.
We shall focus on the piece of cohomology encoded combinatorially in the boundary. As explained in [BMV11], the moduli spaces Ag admit several well-known toroidal com- pactifications, including the perfect cone and second Voronoi compactifications [AMRT75]. The boundary complexes of these compactifications, which record the irreducible compo- nents of the boundary and the combinatorics of how they intersect, may be interpreted as tropical moduli spaces of principally polarized abelian varieties. Moreover, understanding the rational homology of these boundary complexes would yield new results on the ratio- nal cohomology of Ag itself, using the comparison theorems and machinery explained in [CGP18]. We shall understand how these pieces fit together, and then design and imple- ment computer calculations to study the topology of these tropical moduli spaces, thereby producing cohomology calculations of Ag that were previously inaccessible.
The proposed project fits in to a larger program of identifying boundary complexes of compactified moduli spaces with tropical moduli spaces and using this combinatorial intepretation for applications, for example [CMP, CHMR16, CMR16]. The best-developed case so far is the moduli spaces M_g of curves and its Deligne-Mumford compactification, whose boundary complex was identified with the tropical moduli space of curves [ACP15]. This identification was then used in [CGP18, CGP19] to construct previously unknown rational cohomology classes in M_g.
Recommended background:
- Since the project is computational at heart, several group members should have demonstrated strength and interest in computer computations, for example in computer algebra systems such as in Sage or Macaulay2.
- Interest in combinatorial aspects of algebraic geometry; a good indicator is basic familiarity with toric varieties, rational polyhedral cone complexes, and polytopes.
- Familiarity with tropical geometry or moduli spaces is a plus, but not absolutely necessary.
References
[ACP15] D. Abramovich, L. Caporaso, and S. Payne, The tropicalization of the moduli space of curves, Ann. Sci. E ́c. Norm. Sup ́er. (4) 48 (2015), no. 4, 765–809.
[AMRT75] A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmet- ric varieties, Math. Sci. Press, Brookline, Mass., 1975, Lie Groups: History, Frontiers and Applications, Vol. IV. MR 0457437
[BMV11] Silvia Brannetti, Margarida Melo, and Filippo Viviani, On the tropical Torelli map, Adv. Math. 226 (2011), no. 3, 2546–2586. MR 2739784
[Bor74] Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. E ́cole Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 0387496
[CGP18] Melody Chan, Søren Galatius, and Sam Payne, Tropical curves, graph complexes, and top weight cohomology of Mg, arXiv:1805.10186, 2018.
[CGP19] Melody Chan, Søren Galatius, and Sam Payne, Topology of moduli spaces of tropical curves with marked points, arXiv:1903.07187, 2019.
[CHMR16] Renzo Cavalieri, Simon Hampe, Hannah Markwig, and Dhruv Ranganathan, Moduli spaces of rational weighted stable curves and tropical geometry, Forum Math. Sigma 4 (2016), e9, 35. MR 3507917
[CMP] Lucia Caporaso, Margarida Melo, and Marco Pacini, Tropicalizing the moduli space of spin curves, arXiv:1902.07803.
[CMR16] Renzo Cavalieri, Hannah Markwig, and Dhruv Ranganathan, Tropicalizing the space of admissible covers, Math. Ann. 364 (2016), no. 3-4, 1275–1313. MR 3466867
[Hai02] Richard Hain, The rational cohomology ring of the moduli space of abelian 3-folds, Math. Res. Lett. 9 (2002), no. 4, 473–491. MR 1928867
[Har71] G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. E ́cole Norm. Sup. (4) 4 (1971), 409–455. MR 0309145
Project 3: Moduli Spaces of Curves with Cyclic Action
Leadership: Emily Clader (San Francisco State University) & Rohini Ramadas (Brown University)
The moduli space M_{0,n}-bar of genus-zero curves with n distinct marked points is a fundamental object in algebraic geometry. This is in part due to its applicability — the moduli space of curves is connected to such fields as enumerative geometry, combinatorics, and mathematical physics — but also, it is interesting as a variety in its own right. It occupies an intriguing middle ground between the simplicity of toric varieties and the intractability of a more general setting. For example, although M_{0,n}-bar is not toric for n greater than 4, it enjoys some of the same structure that a toric variety would have, such as an explicitly-presented cohomology ring that is combinatorially describable in terms of boundary strata.
One way in which to probe the connection between M_{0,n} and toric varieties is to modify the moduli problem so that a toric moduli space is indeed obtained. This was first carried out by Losev and Manin, who defined a space L_n that parametrizes genus-zero curves with two distinct marked points and n possibly-coinciding marked points, under an appropriate stability condition. The variety L_n is toric, and moreover, its torus-invariant strata are precisely the boundary strata of the moduli space. Combinatorially, these strata label the faces of the (n-1)-dimensional permutahedron, which is the polytope for the toric variety L_n.
Losev and Manin's work was generalized by Batyrev and Blume from the perspective of root systems. There is a construction of a toric variety associated to any root system, and Batyrev--Blume proved that L_n is the toric variety associated to the classical root system A_{n-1}. They then gave an analogous modular interpretation of the toric variety associated to the root system B_n: it parametrizes genus-zero curves with an involution sigma, one marked fixed point of sigma, one marked orbit of sigma, and n additional marked possibly-coinciding orbits of sigma. The moduli space of such objects is not only toric, but, as in the Losev--Manin case, its torus-invariant strata are precisely the boundary strata. The associated polytope is a signed permutahedron.
While Batyrev--Blume have explored the generalizations of this construction to other root systems (and found that, in types C and D, the toric varieties do not have an equally well-behaved modular interpretation), the goal of this project is to pursue a different generalization of both Losev--Manin and Batyrev--Blume's work that is both modular and combinatorially rich.
Consider a moduli space M_n^r that parametrizes genus-zero curves C with following additional data:
- an action sigma of Z/rZ on C;
- a marked fixed point of sigma;
- a marked orbit of sigma;
- n marked possibly-coinciding orbits of sigma.
The first goal of the project will be to carefully prove that a fine moduli space M_n^r indeed exists, and to identify it explicitly for small values of n and r. From here, we can ask: is M_n^r toric? This is the case when r=1 or r=2, but it is likely false in general, and a second goal is to construct at least one example of a moduli space M_n^r that is not toric. Analogously to the classical moduli space of curves, though, we expect M_n^r to have an interesting combinatorial structure even when it is not toric; indeed, this is the reason for our interest in this moduli space. Ambitious project goals might be to understand the combinatorics of the boundary of M_n^r — if not in terms of polytopes (as in the toric case) then in terms of some other combinatorial structure — and to give an explicit presentation of this moduli space's cohomology.
Preferred Background: Good working knowledge of the moduli space of curves, along the lines of Kock and Vainsencher’s book “An invitation to quantum cohomology".
Project 4: Tropical degree-two del Pezzo surfaces and their 56 lines
Leadership: María Angélica Cueto (Ohio State University) & Cecilia Salgado (UFRJ)
A persistent challenge in tropical geometry is to emulate tropical versions of classical results in algebraic geometry. One classical example that is absent from the tropical literature, due to its computational complexity, is the case of degree two del Pezzo surfaces and their moduli. In particular, the arrangement of 56 lines on a the classical surfaces becomes an arrangement of 56 metric trees on the corresponding tropical surfaces. Their combinatorial and metric structure remain a mystery.
In addition to their well-known interpretation of these surfaces as blowups of the projective plane at seven generic points, they also arise as double-covers of the plane ramified over a smooth quartic curve $C$. The 56 lines on the surface come in $28$ pairs, one pair for each bitangent line to the curve $C$. Recent work on tropical moduli spaces of del Pezzo surfaces and tropical bitangents to smooth plane quartics has provided tools to deal with del Pezzo surfaces of degree two in the tropical realm. The goal of this project is to explicitly compute the tropical moduli spaces of degree-two del Pezzos, describe their combinatorial types and explore the interplay between the arrangement of metric trees in the surface and the bitangent lines to tropical quartic curves.
Background: Familiarity with del Pezzo surfaces and tropicalization would be helpful. Fondness for combinatorial methods in Algebraic Geometry would be essential. The project will rely on heavy computations using Sage.
References
[1] M. Baker, Y. Len, R. Morrison, N. Pflueger, and Q. Ren. Bitangents of tropical plane quartic curves. Math. Z., 282(3-4):1017–1031, 2016.
[2] V. V. Batyrev and O. N. Popov. The Cox ring of a del Pezzo surface. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), volume 226 of Progr. Math., pages 85–103. Birkhäuser Boston, Boston, MA, 2004.
[3] M. A. Cueto and A. Deopurkar. Anticanonical tropical cubic del pezzos contain exactly 27 lines. arXiv:1906.08196, 2019.
[4] I. V. Dolgachev. Classical algebraic geometry: a modern view. Cambridge University Press, Cambridge, 2012.
[5] M. Kuwata. Twenty-eight double tangent lines of a plane quartic curve with an involution and the Mordell-Weil lattices. Comment. Math. Univ. St. Pauli, 54(1):17–32, 2005.
[6] D. Maclagan and B. Sturmfels. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015.
[7] Y. I. Manin. Cubic forms, volume 4 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1986. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel.
[8] M. Pieropan. On the unirationality of Del Pezzo surfaces over an arbitrary field. Master’s thesis, Concordia University, 2012.
[9] Q. Ren, K. Shaw, and B. Sturmfels. Tropicalization of del Pezzo surfaces. Adv. Math., 300:156–189, 2016.
[10] C. Salgado, D. Testa, and A. Várilly-Alvarado. On the unirationality of del Pezzo surfaces of degree 2. J. Lond. Math. Soc. (2), 90(1):121–139, 2014.
[11] M. Stillman, D. Testa, and M. Velasco. Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces. J. Algebra, 316(2):777–801, 2007.
[12] D. Testa, A. Várilly-Alvarado, and M. Velasco. Cox rings of degree one del Pezzo surfaces. Algebra Number Theory, 3(7):729–761, 2009.
[13] D. Testa, A. Várilly-Alvarado, and M. Velasco. Big rational surfaces. Math. Ann., 351(1):95–107, 2011.
Project 5: Basepoint free cycles on the moduli space of genus zero curves
Leadership: Angela Gibney (Rutgers University, New Brunswick) & Linda Chen Swarthmore College)
Geometric information about projective varieties can be obtained by studying basepoint free cycles. Basepoint free divisors are the pullbacks of ample divisors along morphisms, and as such we think of them as giving rise to morphisms to other projective varieties. Similarly, basepoint free cycles of higher codimension are also pullbacks of loci along morphisms. We study the basepoint free loci of all codimension on the moduli space of pointed genus zero curves that come from the Gromov-Witten theory of Grassmannians and other homogeneous spaces. The aim of this project is to calculate these loci and gather data to explore several questions: finding conditions that guarantee that such loci are nontrivial, determining which are extremal in nef cones, and finding relations between these and other basepoint free classes.
Preferred Background: Basics of moduli spaces of curves, Gromov-Witten theory, and Grassmannians.
Project 6: A hands-on introduction to aspects of the minimal model program and to Noether-Lefshetz problems
Leadership: Antonella Grassi (University of Pennsylvania) & Julie Rana (Lawrence University)
The objects of the proposed study are threefolds with negative Kodaira dimension, in particular toric varieties.The applications are density and codimension results for Noether-Lefschetz loci in these singular varieties.
The classical Noether–Lefschetz theorem states that any curve in a very general surface X in P^3 of degree d ≥ 4 is a restriction of a surface in the ambient space. The Noether–Lefschetz locus is the locus of surfaces in P^3 of fixed degree where the Picard number is greater than 1.
Of particular interest are the strata of maximal and minimal codimension components.
The surfaces in these strata are characterized by classical geometrical properties.
These results have been partially extended to other varieties.
The project is to use both theory and computational software, such as Macaulay2, to characterize singular (polarized) varieties for which the classical Noether-Lefshetz type theorems hold. The needed properties include vanishings and global generation, both of which are crucial ingredients in the Minimal Model Program. If there is interest we will discuss applications to curve counting invariants.
Preferred Background: Working knowledge of toric geometry at the level of Cox’s MSRI lectures. (see also Cox, Little, Schenck. Toric varieties, vol. 124 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2011)
References
G. Blekherman, J. Hauenstein, J. C. Ottem, K. Ranestad, and B. Sturmfels, Algebraic boundaries of Hilbert’s SOS cones, Compos. Math., 148 (2012), pp. 1717–1735.
U. Bruzzo, A. Grassi, and A. Lopez. Existence and density of general components of the Noether Lefschetz locus on normal threefolds. arXiv:1706.02081, 2017.
M. L. Green Components of maximal dimension in the Noether-Lefschetz locus, J. Diff. Geom., 29 (1989), pp. 295–302.
D. Maulik and R. Pandharipande. Gromov-Witten theory and Noether-Lefschetz theory. In A celebration of algebraic geometry, volume 18 of Clay Math. Proc., pages 469–507. Amer. Math. Soc., Providence, RI, 2013.
C. Voisin, Composantes de petite codimension du lieu de Noether-Lefschetz, Comment. Math. Helv., 64 (1989), pp. 515–526.
Project 7: Hyper-Kähler manifolds, moduli spaces of sheaves, and Lagrangian fibrations
Leadership: Giulia Saccà (Columbia University) & Chiara Camere (University of Milano)
The role of hyper-Kähler (HK) manifolds in algebraic geometry has grown very much in recent years. In this respect two important items, both for studying and constructing examples, are moduli spaces of sheaves on K3 surfaces and Lagrangian fibrations. There will be two proposed projects, both centered around these two structures. In the first one, the participants will construct examples of (possibly singular) Lagrangian fibrations, fibered in Prym varieties. In the second project, the participants will compute the Mordell-Weil group (i.e., the group of rational sections), of some HK manifolds fibered in Jacobians of curves.
Preferred Background, Any of the following: Some background on coherent sheaves (mostly: on curves and smooth projective surfaces); Jacobians (and compactified Jacobians) of curves; curves and linear systems on surfaces; Some background on HK Manifolds
Project 8: Automorphisms of K3 surfaces
Leadership: Alessandra Sarti (Université de Poitiers) and Paola Comparin (UFRO Universidad de La Frontera)
The main objects of the project are K3 surfaces and their automorphisms. In general, the group of automorphisms plays an important role if one wants to understand the geometric properties of a variety. On the other hand, the K3 surfaces are beautiful algebraic surfaces, with several interesting properties. The easiest example of a K3 surface is the Fermat quartic surface in 3-dimensional complex projective space. One remarkable property of K3 surfaces is that the second cohomology group with integer coefficients has the structure of a lattice, i.e. of a free module over the integers with a non-degenerate bilinear form. By using strong results from lattice theory, one can show several fundamental properties of K3 surfaces (Torelli theorem, surjectivity of the period map). The lattice theory is also an important tool when studying the automorphisms group. The aim of the project will be to classify automorphisms of K3 surfaces and/or to study an interesting class of examples which are K3 surfaces carrying an elliptic fibration: one could for example ask about the possible automorphisms of such K3 surfaces and how to describe them. For the project, the computer program MAGMA could be useful.
Preferred background: Basic knowledge of algebraic geometry, as in Hartshorne’s book. A more advanced knowledge of algebraic surfaces and in particular on K3 surfaces is useful, but not necessary.
References
W. P. Barth; K. Hulek; Ch. A. M. Peters; A. Van de Ven. Compact complex surfaces.Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 4. Springer-Verlag, Berlin, 2004.
A. Beauville. Complex algebraic surfaces. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. Second edition. London Mathematical Society Student Texts, 34. Cambridge University Press, Cambridge, 1996.
Project 9: Invariants of Fano threefolds over nonclosed fields
Leadership: Isabel Vogt (Stanford University) & Bianca Viray (University of Washington)
Over algebraically closed fields, all Fano curves (i.e., genus 0 curves) and Fano surfaces (i.e., del Pezzo surfaces) are rational; however, they may fail to be rational over nonclosed fields. For example, they may fail to have rational points or they may have nontrivial Brauer groups. Beginning in dimension 3, the situation is more subtle. Fano threefolds have been classified, but not all Fano varieties are even geometrically rational. In this project we will study invariants of some classes of Fano threefolds over non-algebraically closed fields.
Applicants should have experience with, or at least interest in, working over non-algebraically closed fields. Knowledge of other topics including Brauer groups and higher-dimensional algebraic geometry would be helpful.
Publications
- Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, Corey Wolfe, On the Top-Weight Rational Cohomology of \$\mathcal\A\_g\$., arXiv: Algebraic Geometry (2020).
- Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon, Libby Taylor, Nivedita Viswanathan, Higher Fano Manifolds, 2021.
- Michela Artebani, Alessandra Sarti, Non-symplectic automorphisms of order 3 on K3 surfaces, Math. Ann. 342 (2008), 903-921.
- Sarah Frei, Lena Ji, Soumya Sankar, Bianca Viray, Isabel Vogt, Curve classes on conic bundle threefolds and applications to rationality, arXiv preprint arXiv:2207.07093 (2022).