### Juliette Bruce

Broadly, I am interested in commutative algebra and algebraic geometry, as well as their interactions with other fields like number theory and topology. My specific research interests are a bit widespread, really I am driven by interesting and fun sounding questions. That said, currently, I spend most of my time thinking about free resolutions, syzygies, and the geometry of the Hilbert scheme of points.

Here is a list of my publications:

1. Asymptotic Syzygies in the Setting of Semi-Ample Growth
pre-print.
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.

2. Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties
submitted.
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 $$E^n$$

3. 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.

4. 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.

5. A probabilistic approach to systems of parameters and Noether normalization
submitted.
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.

6. 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.

7. 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.