This paper uses the same methods as my first paper as a graduate student, Collections of hypersurfaces containing a curve to prove that, in the space of all r-tuples of homogenous forms of some fixed degrees, a component of maximal dimension of the locus of tuples of forms with positive dimensional common vanishing locus in P^r either consists of forms all vanishing on some line or forms where a proper subset fail to form a complete intersection.

This project was from the Arizona Winter School. A¹-enumerative geometry provides an arithmetic enrichment to the usual notion of degree, where the A¹-degree takes values in the Grothendieck-Witt ring of the base field. We compute A¹-degrees of finite maps between affine spaces induced by actions of Weyl groups. By using the algorithm to compute A¹-degree already in the literature, the problem quickly reduced to verifying some facts about flag varieties and Weyl groups.

Chang and Ran bounded the slope of M₁₅ from below to show M₁₅ has Kodaira dimension -∞. They subsequently used the result on M₁₅ to deduce the corresponding result on M₁₆. In this note, we correct the formula used to compute chern numbers of degeneracy loci and correct the computation of Chang and Ran. We show that their argument still shows that M₁₅ has Kodaira dimension -∞, but it is insufficient to show M₁₆ has Kodaira dimension -∞. In particular, the question of whether M₁₆ is uniruled is still open.

Given a plane curve C, there is an associated GL₃ -orbit closure in the P^N of all plane curves. Our goal is to degenerate such an orbit by specializing the underlying curve C and determine what other "sibling" orbits appear. We determine this when C acquires nodes and cusps, and when C splits off lines as components. In the case of quartics, the degeneration to the double conic is particularly nice and allows us to compute equivariant classes of orbit closures of many quartic plane curves. As an enumerative consequence, we see that a general quartic threefold has 510720 2-plane sections yielding a fixed, general genus 3 curve.

For a partition P of a positive integer d, we have a subvariety of Sym^dA¹ parameterizing (the closure of the) d-tuples of unordered points on A¹ with multiplicities given by P. It is known that this construction can be generalized to complex multiplicities as long as no subset of the complex weights add to zero, and varying the complex weights gives a family of varieties over the open subset of affine space of allowable weights.

Our goal is to expand on this construction of families of incidence strata with complex weights on A¹. We first show the construction extends to any affine variety. We give bounds on the embedding dimension of the family for a small number of complex weights. In the case of an affine curve, we determine the singular locus of a stratum by understanding its normalization. Finally, following a suggestion of Etingof, we give an equivalent definition using complex tensor powers in Deligne categories.

The first counterexample to the Slope Conjecture on effective divisors on the moduli space of curves was given by Farkas and Popa, where they used the divisor in M₁₀ consisting of genus 10 curves that can be embedded into a quadric hypersurface in P⁴ as a degree 12 curve. This has since been generalized to curves lying in a quadric of low rank by Farkas and Rimanyi. This preprint generalizes this to all divisorial conditions on the hypersurfaces of a fixed degree containing a curve.

However, there are a few caveats:

• The counterexamples we find are all virtual, meaning one still needs to check their support is not the entire moduli space.
• The calculation of the slopes does not include the coefficients of the boundary classes δ_i for i>0.
• Like previous papers using the same method, we only consider projective embeddings with Brill-Noether number zero.

We consider PGL₂ -equivariant classes of strata of points in the projective line corresponding to when subsets of the points come together. In the case of n ordered points on a line, we show that the PGL₂ equivariant Chow ring of (P¹)ⁿ has a particularly pleasant form. For n at least 3, it is generated by the divisors corresponding to when two of the n points come together and the relations are generated by two types of geometrically obvious relations. We also find additive bases for the equivariant Chow ring in terms of these strata.

For the case n unordered points on a line, parameterized by SymⁿP¹, we extend the computation of GL₂-equivariant classes of these coincident root loci given by Feher, Nemethi, and Rimanyi (2006) to PGL₂-equivariant classes. We show that nontrivial 2-torsion does occur in very special cases, but regardless linear relations between these classes that hold rationally also hold integrally.

Given n points spanning projective d-space, we can naturally associate to it point configurations in projective r-space for all r (possibly larger, equal to, or smaller than d) by taking images under all linear rational maps from P^d to P^r. These give GL_r+1-equivariant loci in (P^r)ⁿ and are GL_r+1-orbits for r at least d.

• We compute the classes of these loci in GL_r+1-equivariant cohomology.
• We show that a formal sum amalgamating these classes over all choices of r satisfies recursive properties analogous to the quantum product.
• Degeneration of these loci gives relations among matroid polytopes of point configurations. We show that conversely every such relation among matroid polytopes is given by degeneration.

Using the case of n points on a line, we can answer the question of how many line sections of a hypersurface have a fixed moduli, generalizing a computation by Cadman and Laza (2008)

Chang and Ran showed that the largest complete family of immersed curves in projective n-space has dimension n-2, provided the the curves are of positive genus. We show their methods extend to the case of genus zero. We also note that their methods also show complete families of immersed curves in n-dimensional rational homogenous varieties have dimension at most n-2.

Riedl and Yang (2016) showed that the Kontsevich space of rational curves on Fano hypersurfaces of Fano index at least 3 is irreducible and of the expected dimension. We use their methods combined with results from the previous paper Collections of hypersurfaces containing a curve to extend their result to the case of low degree rational curves on Fano hypersurfaces of Fano index 2. We show there is exactly two components of the Kontsevich space, where one component consists of multiple covers of a line.

The problem considered in the previous paper Collections of hypersurfaces containing a curve can be regarded as a special case of the following more general problem: Given a globally-generated vector bundle V on a projective variety X, what is the largest component in the affine space of sections H⁰(V) consisting of sections with zero locus that is larger dimensional than expected? In general, this problem seems hopeless, but after replacing V with a large twist V(N) for N>>0, we show that the largest component consists of sections vanishing on a subvarieties of X that have minimal degree among subvarieties of dimension dim(X)-rank(V)+1.

Given a list of degrees (d₁,...,d_n), we have the vector space parameterizing n-tuples of homogenous forms in n+1 variables with those degrees. Inside that vector space, there is a bad locus of n-tuples of forms whose zero locus in projective n-space is positive dimensional. In this paper, we show that if the degrees are not too far apart, then the unique largest component consists of forms that vanish along some line. We give two applications:

• We identify the largest component(s) of the locus of smooth hypersurfaces containing a larger dimensional family of lines through some point than expected.
• We also improve on Slavov's work (2015) regarding the largest component of the locus of hypersurfaces with a positive-dimensional singular locus.

A topological covering map between finite graphs induces a surjection from the critical group of the covering space to the critical group of the base space. In the case of regular coverings, we give an interpretation for the kernel of this map as a voltage graph critical group. In the special case of double coverings, we use this notion to interpret Bai's computation (2003) of the critical group of the hypercube away from 2-torsion.

We positively resolve the Splitting Subspace Conjecture, stated by Ghorpade and Ram and stemming from a question of Niederreiter (1995). The question is: given a generator σ of the finite field extension F_q^mn over F_q, how many m-dimensional F_q-vector subspaces W of F_q^mn satisfy the property that W, σW,...,σ^n-1W span F_q^mn? We extend the problem to a wider class of counting problems and develop a recursion to solve this wider class all at once. Dennis Stanton found the closed form solution to the recursion but generously declined to be listed as a coauthor.

Applying the greedy algorithm to generate a sequence of nonnegative integers satisfying the property that there are no distinct x, y, and z in the sequence such that x+y=2z yields the nonnegative integers that have no 2's in their base 3 representation. Layman (1999) generalized this description to the equation x+y+z=3w. Here, we generalize this further, in particular finding a closed form expression for the greedy sequence avoiding solutions to the equation x₁+...+x_n=nx_n+1 for distinct x₁,...,x_n+1.