### Juliette Bruce

My research interests lie in algebraic geometry, commutative algebra, and arithmetic geometry. In particular, I am interested in using homological methods to study the geometry of zero loci of systems of polynomials (i.e. algebraic varieties). I am also interested in studying the arithmetic properties of varieties over finite fields. Further, I am passionate about promoting inclusivity, diversity, and justice in math.

#### Publications

1. The Quantitative Behavior of Asymptotic Syzygies for Hirzebruch Surfaces. Journal of Commutative Algebra, to appear.
The goal of this note is to quantitatively study the behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $$\mathcal{O}(d,2)$$ conform to Ein, Erman, and Lazarsfeld's normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.
2. The Virtual Resolutions Package for Macaulay2. Journal of Software for Algebra and Geometry, to appear. with Ayah Almousa, Michael C. Loper, Mahrud Sayrafi
We introduce the VirtualResolution package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in $$\mathbb{P}^1\times\mathbb{P}^2$$, providing sources for interesting virtual resolutions.
3. A probabilistic approach to systems of parameters and Noether normalization. Algebra and Number Theory, 13 (2019), no. 9, 2081-2102. with Daniel Erman
We study systems of parameters over finite fields from a probabilistic perspective. Our central technique is an adaptation of Poonen's closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. Applications include an effective Noether normalization result over a finite field, and a new proof of a recent result of Gabber-Liu-Lorenzini and Chinburg-Moret-Bailly-Pappas-Taylor on uniform Noether normalizations for projective families over the integers. Last updated: 9/21/17, 20 pages.
4. Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties. Proc. Amer. Math. Soc., 148 (2020), no.2, 535-551. with Wanlin Li
We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, we show that for simple abelian varieties a better bound is possible. As an application of these results we show that if $$E$$ is an elliptic curve over a finite field then for any $$n\in\mathbb{N}$$ there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to $$E^n$$
5. Conjectures and computations about Veronese syzygies. Experimental Mathematics, to appear. with Daniel Erman, Steve Goldstein, Jay Yang
We formulate several conjectures which shed light on the structure of Veronese syzygies of projective spaces. Our conjectures are based on experimental data that we derived by developing a numerical linear algebra and distributed computation technique for computing and synthesizing new cases of Veronese embeddings for $$\mathbb{P}^2$$. Last updated: 11/9/17, 25 pages.
6. The degree of SO(n). Combinatorial Algebraic Geometry, 207-224, Fields Inst. Commun. 80, (2017). with Madeline Brandt, Taylor Brysiewicz, Robert Krone, Elina Robeva
We provide a closed formula for the degree of $$\text{SO}(n)$$ over an algebraically closed field of characteristic zero. In addition, we describe symbolic and numerical techniques which can also be used to compute the degree of $$\text{SO}(n)$$ for small values of $$n$$. As an application of our results, we give a formula for the number of critical points of a low-rank semidefinite programming optimization problem. Finally, we provide some evidence for a conjecture regarding the real locus of $$\text{SO}(n)$$. Last updated: 1/13/17, 21 pages.
7. Monomial valuations, cusp singularities, and continued fractions. Journal of Commutative Algebra, 7 (2015), no. 4, 495-522. with Molly Logue, Robert Walker
This paper explores the relationship between real valued monomial valuations on $$k(x,y)$$, the resolution of cusp singularities, and continued fractions. It is shown that up to equivalence there is a one to one correspondence between real valued monomial valuations on $$k(x,y)$$ and continued fraction expansions of real numbers between zero and one. This relationship with continued fractions is then used to provide a characterization of the valuation rings for real valued monomial valuations on $$k(x,y)$$. In the case when the monomial valuation is equivalent to an integral monomial valuation, we exhibit explicit generators of the valuation rings. Finally, we demonstrate that if v is a monomial valuation such that $$\nu(x)=a$$ and $$\nu(y)=b$$, where $$a$$ and $$b$$ are relatively prime positive integers larger than one, then $$\nu$$ governs a resolution of the singularities of the plane curve $$x^b=y^a$$ in a way we make explicit. Further, we provide an exact bound on the number of blow ups needed to resolve singularities in terms of the continued fraction of $$a/b$$. Last updated: 1/14/16, 28 pages.
8. Betti tables of reducible algebraic curves. Proc. Amer. Math. Soc. 142 (2014), 4039-4051. with Pin-Hung Kao, Evan D. Nash, Ben Perez, Pete Vermeire
We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses. We also give explicit formulas for the entries of the Betti tables for all curves of genus zero and one. Last, we give formulas for the graded Betti numbers for a class of curves of higher genus. Last updated: 2/23/13, 13 pages.

#### Pre-Prints

1. Characterizing Multigraded Regularity on Products of Projective Spaces pre-print. with Lauren Cranton Heller, Mahrud Sayrafi
We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions. We prove that on a product of projective spaces $$X$$, the multigraded regularity region of a module $$M$$ is determined by the minimal graded free resolutions of the truncations $$M_{\geq\textbf{d}}$$$for $$\textbf{d}\in\text{Pic} X$$. Further, by relating the minimal graded free resolutions of $$M$$ and $$M_{\geq\textbf{d}}$$ we provide a new bound on multigraded regularity of $$M$$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections. 2. Syzygies of $$\mathbb{P}^{1}\times\mathbb{P}^{1}$$: Data and Conjectures pre-print. with Daniel Corey, Daniel Erman, Steve Goldstein, Robert P. Laudone, Jay Yang We provide a number of new conjectures and questions concerning the syzygies of $$\mathbb{P}^{1}\times\mathbb{P}^{1}$$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of $$\mathbb{P}^{1}\times\mathbb{P}^{1}$$. These computations utilize linear algebra over finite fields and high-performance computing. 3. On the top-weight rational cohomology of $$\mathcal{A}_g$$ pre-print. with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, Corey Wolfe We compute the top-weight rational cohomology of $$\mathcal{A}_g$$ for $$g=5$$, $$6$$, and $$7$$, and we give some vanishing results for the top-weight rational cohomology of $$\mathcal{A}_{8}, \mathcal{A}_{9}$$ and $$\mathcal{A}_{10}$$. When $$g=5$$ and $$g=7$$, we exhibit nonzero cohomology groups of $$\mathcal{A}_g$$ in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of $$\mathcal{A}_g$$ and the homology of the link of the moduli space of principally polarized tropical abelian varieties of rank $$g$$. To compute the latter we use the Voronoi complexes used by Elbaz-Vincent-Gangl-Soul\'e. Our computations give natural candidates for compactly supported cohomology classes of $$\mathcal{A}_g$$ in weight$0\$ that produce the stable cohomology classes of the Satake compactification of $$\mathcal{A}_g$$ in weight $$0$$, under the Gysin spectral sequence for the latter space.
4. The SchurVeronese package in Macaulay2. submitted. with Daniel Erman, Steve Goldstein, Jay Yang
This note introduces the Macaulay2 package SchurVeronese, which gathers together data about Veronese syzygies and makes it readily accessible in Macaulay2. In addition to standard Betti tables, the package includes information about the Schur decompositions of the various spaces of syzygies. The package also includes a number of functions useful for manipulating and studying this data.
5. Asymptotic Syzygies in the Setting of Semi-Ample Growth. submitted.
We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of $$\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}$$ embedded by $$\mathcal{O}_{\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}}(d_1,d_2)$$ are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.