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.

When r=1, this recovers the Losev--Manin space L_n, while when r=2, it recovers the Batyrev--Blume space. For r greater than 2, it has not yet been studied, and this is the work that we propose to undertake.
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".

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.

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.

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.

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

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.

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.