TAGS 2014

Texas Algebraic Geometry Conference Posters

On Real Rationally Connected Varieties, Derek Allums (Rice University)

One of the reasons projective space is easy to work with is that it contains many rational curves. Seeking a generalization in this direction, a variety $X$ is said to be rationally connected if, roughly, it also has a lot of rational curves. This turns out to be an extremely useful property. We review some classical results and give some new results concerning the existence of non-constant morphisms of curves $C$ to a rationally connected variety $X$. In particular, under very mild hypotheses, we show such a map always exists.

Non-rational varieties of adjoint algebraic groups, Nivedita Bhaskhar (Emory University)

A k-variety is said to be rational if its function field is purely transcendental over k. The first example of a non-rational adjoint k-group PSO(q) was given by Merkurjev as a consequence of his computations of R-equivalence classes of adjoint classical groups. The quadratic form in question has non-trivial discriminant which property is used crucially in the proof. Gille provided the first example of a quadratic form of trivial discriminant whose associated adjoint group is non-rational. We give a recursive construction to produce examples of $k_n$-quadratic forms $q_n$ in the n-th power of the fundamental ideal in the Witt ring whose corresponding adjoint groups PSO($q_n$) are not (stably) rational.

Andean component of codimension two lattice basis ideals, Zekiye Eser (Texas A&M University), Laura F. Matusevich (Texas A&M University)

We provide explicit combinatorial descriptions of the primary components of codimension two lattice basis ideals. The components of these binomial ideals can be computed via adding certain monomials and using saturation. This is joint work with Laura F. Matusevich.

On the moduli space of quintic surfaces, Patricio Gallardo (Stony Brook University)

We describe the use of GIT and stable replacement for studying the geometry of a special compactification of the moduli space of smooth quintic surfaces, the KSBA compactification. In particular we discuss the interplay between non-log-canonical singularities and boundary divisors. This interplay generalizes similar phenomena on the moduli space of curves of genus three.

Complexity of matrix-vector multiplication and matrix rigidity, Fulvio Gesmundo (Texas A&M University), J. Hauenstein (NCSU), C. Ikenmeyer (TAMU), JM Landsberg (TAMU)

A matrix A is said to be (s,r)-non-rigid if one can change s entries of A such that the new matrix has rank at most r; it turns out that this notion has a strong connection with the complexity of matrix-vector multiplication. In recent work with J. Hauenstein, C. Ikenmeyer and J.M. Landsberg, we provide general results about the defining ideal of some particular joins of varieties, allowing us to exhibit many non-obvious equations testing for (border) rigidity; moreover classical tools in algebraic geometry lead to results concerning the dimension and the degree of such joins.

16051 Formulas for Ottaviani's Invariant of Cubic Threefolds, Christian Ikenmeyer (Texas A&M University), Abdelmalek Abdesselam (University of Virginia), Gordon Royle (University of Western Australia)

We provide explicit combinatorial formulas for Ottaviani's degree 15 invariant which detects cubics in 5 variables that are sums of 7 cubes. Our approach is based on the chromatic properties of certain graphs and relies on computer searches and calculations.

Intermediate Jacobian of toric surface fibrations over curves, Nikita Kozin (Rice University)

Given a threefold which fibers into toric surfaces, there is a natural way to realize its Intermediate Jacobian as a Prym variety. From the other side Neron-Severi torus of the generic fiber can be used to realize it as a component of space of torsors under the action of its Neron model. The poster explains my progress on relating above two constructions.

Tropical Theta Functions, Travis Mandel (University of Texas at Austin)

In toric geometry, one uses the combinatorics of a character lattice and its dual to study the toric varieties. Gross, Hacking, and Keel (GHK) have used ideas from mirror symmetry to generalize this setup log Calabi-Yau surfaces which are not necessarily toric. The analogs of the character lattice and its dual are no longer lattices, but by looking at the tropicalizaitons of GHK's theta functions I can still describe a canonical pairing between these two spaces (generalizing the dual pairing). This allows several useful constructions from toric geometry (dual cones, polar polytopes, etc.) to be generalized to the log Calabi-Yau situation.

Dynamic alpha-invariants of del Pezzo surfaces, Jesus Martinez-Garcia (Johns Hopkins University), I. Cheltsov (University of Edinburgh)

Tian's alpha-invariant of a Fano variety X of dimension n can be generalised for log smooth pairs (X,D) where D is an anticanonical divisor, to obtain a dynamic alpha-invariant. A Theorem of Jeffres, Mazzeo and Rubinstein tells us that if the alpha-invariant of (X, (1-b)D) is bigger than n/(n+1), then X admits a Kahler-Einstein metric with edge singularities along D of angle 2*pi*b. This is always the case for small positive b. Donaldson posed the question of which is the supremum R(X,D) of all b's for which such metrics exist. This is unknown in non-trivial cases. We provide a lower bound for R(X,D) for all smooth surfaces by computing the alpha-invariant.

Generalized Bogomolov-Gieseker Inequality for the Quadric Threefold, Benjamin Schmidt (The Ohio State University)

Bridgeland stability conditions are an adaptation of the concept of slope stability to triangulated categories. The main result on this poster is a certain generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macr\`i and Toda for the case of the smooth quadric threefold. This implies the existence of a large family of Bridgeland stability conditions.

The distribution of S-integral points on SL_2 orbits closures of binary forms, Sho Tanimoto (Rice University), James Tanis (Rice University)

We study the distribution of S-integral points on SL_2 orbit closures of binary forms and extend the result of Duke, Rudnick, and Sarnak. Main ingredients of the proof are the method of mixing developed by Eskin-McMullen and Benoist-Oh, Chambert-Loir-Tschinkel`s study of asymptotic volume of height balls, and Hassett-Tschinkel`s description of log resolutions of SL_2 orbit closures of binary forms in terms of moduli spaces of stable maps.

Gaffney-Lazarsfeld Theorem for Homogeneous Spaces, Yu-Chao Tu, (Princeton University)

We generalize the Gaffney-Lazarsfeld theorem on higher ramification loci of branched coverings of projective spaces to homogeneous spaces with Picard number one.

Double Transitivity of Galois Groups in Schubert Calculus of Grassmannians, Jacob White (Texas A&M University)

We investigate double transitivity of Galois groups in the classical Schubert calculus on Grassmannians. We show that all Schubert problems on Grassmannians of 2- and 3-planes have doubly transitive Galois groups, as do all Schubert problems involving only special Schubert conditions. We also investigate the Galois group of every Schubert problem on Gr(4,8), finding that each Galois group either contains the alternating group or is an imprimitive permutation group and therefore fails to be doubly transitive.

A Hasse principle on hermitian forms, Zhengyao Wu (Emory University)

Let F be a function field of one variable over a complete discrete valued field with residue characteristic not 2. For each discrete valuation v on F, let F_v be the completion of F at v. Let D be a finite-dimensional central division F-algebra with involution \sigma of the first kind. Suppose V is a right D-vector space of dimension n and h: V\times V\to D is a nondegenerate hermitian or skew-hermitian form. This article shows that h has a totally isotropic subspace of reduced dimension $r$ if and only if there is a fixed integer r>0, which is a multiple of \Ind(D), such that for each discrete valuation v there exists a totally isotropic subspace N_v\subset V\otimes_FF_v of reduced dimension r.

Rank Two Brill-Noether Problems, Naizhen Zhang (University of California, Davis)

Higher-rank Brill-Noether problems studies the moduli of vector bundles with certain number of sections on curves. We present some recent progresses in rank two. One focus is to prove existence results for rank two Brill-Noether loci with canonical determinant for an infinite class of new cases. Another one is to discuss the relation between the study of the local geometry of multiply symplectic Grassmannians and proving new expected dimensions for more general Brill-Noether loci in rank two.

Organizing committee:
Brendan Hassett
Anthony Varilly-Alvarado

For administrative or web questions, contact:
Bonnie Hausman