Southwest Local Algebra Meeting

Let \(F\) be a homogeneous polynomial of degree \(n\) in at least \(d^2 +1\) variables over the \(p\)-adic numbers, \(\mathbb{Q}_p\) . Artin conjectured that such \(F\) always have nontrivial zeros in any \(p\)-adic field. Although this has been shown to be false in general, the conjecture is still widely believed to be true for prime degree forms. This conjecture holds for \(d=2\) and \(d=3\) due to Hasse and Lewis, respectively. By the work of Ax and Kochen, the conjecture is also known to hold whenever the characteristic of the residue field is sufficiently large. In this talk, we will explore recent progress for low degree forms towards making bounds on the size of the residue field effective. A wide range of techniques are needed, including Bertini theorems, point counting on curves over finite fields, and computation. This is joint work with Christopher Keyes.

Let \(S\) be a standard graded polynomial ring over a field \(K\). Let \(R = S/I\), where \(I\) is a graded ideal of \(S\). \(R\) is called Koszul if \(K\) has a linear free resolution over \(R\). If \(R\) is Koszul, then \(I\) is generated by linear and quadratic elements, but the converse fails. An ideal \(J\) of \(S\) is linked to \(I\) if there is a complete intersection \(C\) such that \(J = C:I\) and \(I = C:J\). The ideal \(I\) is called licci if there are a finite number of ideals, each linked to the previous by some complete intersection, such that the first ideal is \(I\) and the last is a complete intersection. All such ideals are Cohen-Macaulay.

Huneke, Polini, and Ulrich previously asked whether the licci property of an ideal may be detected from its graded Betti table. While recent examples due to Boocher show that the answer is no, we give a strong positive answer in the case of quadratic ideals. In particular, we show how to identify the Hilbert function of a quadratic licci ideal with a partition of its codimension. Further we give a complete characterization of Koszul licci ideals, in terms of Betti tables, Hilbert functions, and defining equations, showing that they are at most 2 links away from a complete intersection. This is joint work with Paolo Mantero and Matthew Mastroeni.

A "frieze" is an infinite configuration of positive integers satisfying certain determinantal identities. The first examples were studied by Coxeter and later Conway, who showed they were miraculously periodic and counted by the Catalan numbers. In this talk, I will describe friezes whose shape is determined by a tree, and review the recent results on periodicity, finiteness, and enumeration. Remarkably, all of these results are fueled by a connection to cluster algebras, a type of combinatorial commutative algebra which has been an area of explosive research for the last 25 years.

An algebraic variety is a set of zeros of polynomials and naturally equipped with a coordinate ring. Hilbert’s syzygy theorem then asserts that there exists a finite length minimal graded free resolution of the coordinate ring. Information from the resolution can be expressed by a Betti diagram. It has been drawn a great attention to understand algebraic and geometric information in the syzygies of the variety. Modern research can be traced back to Castelnuovo’s study in 1893 on the linear system of algebraic curves. In 1980’s Green developed Koszul homology groups to compute and describe the shape of the Betti diagram. In this talk, we focus ourselves on reviewing recent progress in the study of syzygies of algebraic curves, including a joint work with Jinhyung Park on effective gonality theorem.

Serre defined and studied an intersection multiplicity for finitely generated modules over a regular local ring by using the Euler characteristic, and showed it satisfies many properties that one would expect from an intersection theory. In this talk we discuss a new notion of lifting modules over a noetherian local ring to a regular local ring, and then show how it can be used to prove a new case of Serre's long standing conjecture on the positivity of the Euler characteristic. This is joint work with Nawaj KC, and with Benjamin Katz, Nawaj KC, Kesavan Mohana Sundaram, and Ryan Watson.

Pioneering work of Artin–Tate–Van den Bergh–Zhang extends important aspects of projective geometry to the noncommutative (nc) setting. In particular, the derived category of such a nc scheme shares many features with the derived category of a classical one. In this talk, I'll discuss extensions of some classical and modern results in the theory of nc projective geometry to nc spaces associated to dg-algebras. The focus will be on applications to projective varieties: for instance, this approach results in an analog of a landmark theorem of Orlov concerning the derived category of a complete intersection for any projective variety. The work covered in this talk includes joint work with Michael K. Brown and Andrew Soto Levins.