Contributed Talks

Debjit Basu: Smoothing of Abelian carpets on elliptic ruled surfaces and applications to Moduli

In this talk we study degeneration of polarized Abelian surfaces in projective space where the central fiber is bielliptic. These degenerations give rise to nowhere reduced schemes on elliptic ruled surfaces with invariant e = -1 which have the same numerical invariants as an Abelian surface. These structures are ribbons studied by D. Eisenbud, D. Bayer and M. Green. These led us to study ribbons with same numerical invariants as Abelian surfaces on all elliptic ruled surfaces and we call these ribbons Abelian carpets. The name originates from the earlier works on K3 carpets by F. Schreyer, D. Eisenbud, D. Bayer, B. Purnaprajna, F. J. Gallego, M. Gonzalez, D. Raychaudhuri, J. Mukherjee and our work in Abelian surfaces is inspired by their work for K3 surfaces. Abelian carpets on elliptic ruled surfaces with invariant e = -1 embedded in projective 4-space arose in the investigations of the sections of Horrocks-Mumford bundle by K. Hulek and A. Van de Ven. We study how many non-isomorphic Abelian carpets are there on a elliptic ruled surface and their parameter space, whether all Abelian carpets on elliptic ruled surfaces embedded in any projective space arise as degenerations of polarized abelian surfaces. We also study the nature of the Hilbert scheme parametrizing numerical Abelian surfaces near the point of an Abelian carpet.

Aaron Goodwin: Compactifying the Moduli Space of Marked Plane Curves

Since its development by David Mumford in the 60s, Geometric Invariant Theory has been successfully used to compactify many moduli spaces. Recently, more attention has been paid to the choices made in such compactifications. Combinatorial structures tend to underlie these choices, parametrizing different compactifications and birational equivalences between them. We study GIT compactifications of the moduli space of plane curves with marked points and the wall and chamber decomposition, which describes how weights attached to the curve and the points lead to different compactifications.

Jae Hwang Lee: A Quantum $H^*(T)$-module via Quasimap Invariants

For $X$ a smooth projective variety, the quantum cohomology ring $QH^*(X)$ is a deformation of the usual cohomology ring $H^*(X)$, where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov--Witten invariants. When $X$ is toric with  geometric quotient description $V /\!/T$, the cohomology ring $H^*(V /\!/ T)$ also has the structure of a $H^*(T)$-module. In this paper, we introduce a new deformation of the cohomology of $X$ using quasimap invariants with a light point. This defines a quantum $H^*(T)$-module structure on $H^*(X)$ through a modified version of the WDVV equations. We explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a semipositive toric variety.

Emanuela Marangone: Lefschetz Properties and non-Lefschetz locus

An Artinian graded algebra A has the Weak Lefschetz Property if there is a linear form, $\ell$, such that the multiplication map $\times \ell: A_i \to A_{i+1}$  has maximal rank for each degree. The non-Lefschetz locus is the set of linear forms for which maximal rank fails. Boij--Migliore--Mirò-Roig--Nagel show that, for a general Artinian complete intersection of height 3, the non-Lefschetz locus has the expected codimension.  

In this talk, we also address the same type of question for forms of degree 2 instead of lines. We define the non-Lefschetz locus of conics and show that, for a general complete intersection of height 3, it has the expected codimension as a subscheme of $\mathbb P^5$. 

Finally, we will discuss the geometric meaning of non-Lefschetz lines (and conics), relating them with the splitting lines (and conics, respectively).

Arjun Nigam: Counting lines on the intersection of certain hypersurfaces

There are many celebrated results in enumerative geometry over algebraically closed fields. Perhaps the most famous is the theorem of Salmon and Cayley that there are 27 lines on a smooth cubic over an algebraically closed field. The general strategy for proving these enumerative results is to express the solution set as the zero locus $Z(s)$ of a section $s$ of a vector bundle $V$ over some moduli space $X$. One then shows that the number of points in $Z(s)$ does not depend on $s$ and computes this number. However, when $k$ is not algebraically closed, a naive count of the number of points in $Z(s)$ is not independent of $s$ in general. For example, a smooth cubic surface over the real numbers can have 3, 7, 15, or 27 lines. However, there is an invariant $e(V)$ called the $\mathbb{A}^1$-Euler class of $V$ from which we can deduce the original count in the algebraically closed setting. Under nice orientability assumptions on $V$, a result analogous to the Poincaré–Hopf theorem shows that $e(V)$ can be computed locally using a general section $s$, allowing us to salvage the proof strategy from the classical case. In this talk, I will demonstrate how $e(V)$ can sometimes be locally computed even for non-orientable $V$ by working out the specific moduli problem of counting lines on the intersection of two degree $n-2$ hypersurfaces in $\mathbb{P}^n_k$.

Benjamin Oltsik: Symbolic Defect of Monomial Ideals

Symbolic powers of ideals $I$, denoted $I^{(n)}$, in a Noetherian commutative ring $R$ have long been a topic of interest in commutative algebra.  They are directly related to the associated primes of $I$, and always contain the ordinary powers, $I^n$.  The symbolic defect function is a numerical function designed to measure the ``closeness” of $I^{(n)}$ and $I^n$.  Letting $\mu(M)$ denote the minimal number of generators of a module $M$, symbolic defect, denoted $\operatorname{sdef}_I(n)$, is defined to be $\operatorname{sdef}_I(n) = \mu(I^{(n)}/I^n)$.  In this talk, we will introduce tools to help us study $\mu(I^n), \mu(I^{(n)})$, and $\operatorname{sdef}_I(n)$, particularly for monomial ideals.  We compare the latter function to a related invariant, the integral symbolic defect, and then discuss some recent results.

Christian Wolird: Inversive Sums and the Magic Square of Squares

The puzzle of the "Magic Square of Squares" is to arrange distinct square numbers in a square grid such that the contents of each row, column, and diagonal sum to the same total. The case of a 4x4 grid with integer entries was solved by Euler in 1770. And while other larger grid sizes have been solved in recent decades, the 3x3 case has remained an open problem. In addition to historical overview and geometric interpretation, we will see in this talk that by means of a birational map, there is a correspondence of 3x3 magic squares of squares over Z[i] with certain "inversive sums" over Q(i) (meaning sums fixed by term-wise inversion). Under this correspondence, various "near-miss" solutions have a nice interpretation as affine lines on a relevant complex surface.

Yilong Zhang: An elliptic surface with maximal Picard number

For a smooth algebraic surface over complex numbers, the rank $\rho$ of Neron-Severi group is bounded above by the Hodge number $h^{1,1}$. When $\rho=h^{1,1}$, examples are rare particularly when Kodaira dimension is at least zero. Shioda constructed a sequence of elliptic surfaces with maximal Picard number and has Kodaira dimenions one. Such surface is extremal and has no section of infinite order. In this talk, I will show an example of elliptic surface with the same conditions and with a section of infinite order.