arXiv subjects

Nolan Schock

Publications and source records attributed to Nolan Schock.

5 recordsLinked to original sources

arXiv2023

The $W(E_6)$-invariant birational geometry of the moduli space of marked cubic surfaces

The moduli space $Y = Y(E_6)$ of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth century work of Cayley and Salmon. Modern interest in $Y$ was restored in the 1980s by Naruki's explicit construction of a $W(E_6)$-equivariant smooth projective compactification $\overline{Y}$ of $Y$, and in the 2000s by Hacking, Keel, and Tevelev's construction of the KSBA stable pair compactification $\widetilde{Y}$ of $Y$ as a natural sequence of blowups of $\overline{Y}$. We describe generators for the cones of $W(E_6)$-invariant effective divisors and curves of both $\overline{Y}$ and $\widetilde{Y}$. For Naruki's compactification $\overline{Y}$, we further obtain a complete stable base locus decomposition of the $W(E_6)$-invariant effective cone, and as a consequence find several new $W(E_6)$-equivariant birational models of $\overline{Y}$. Furthermore, we fully describe the log minimal model program for the KSBA compactification $\widetilde{Y}$, with respect to the divisor $K_{\widetilde{Y}} + cB + dE$, where $B$ is the boundary and $E$ is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.

math.AG↗

arXiv2023

Moduli of weighted stable marked cubic surfaces

Let $Y(E_n)$ denote the moduli space of pairs $(S,B)$ where $S$ is a del Pezzo surface of degree $9-n$ and $B$ is the labeled (marked) sum of its finitely many lines. When $n=6$, $Y(E_6)$ is the classical moduli space of marked cubic surfaces dating back to the nineteenth century. We describe the compactifications of $Y(E_5)$ and $Y(E_6)$ by Koll\'ar--Shepherd-Barron--Alexeev (KSBA) weighted stable pairs $(S,cB)$. There is a finite wall-and-chamber decomposition of the weight domain $\left(\frac{9-n}{N},1\right]$, and we explicitly identify this decomposition, as well as describe in detail the weighted stable pairs parameterized by the moduli spaces in each chamber. This generalizes the work of Hacking, Keel, and Tevelev constructing the moduli space and its universal family in the weight 1 case, and in particular yields a complete description of the fibers of this family.

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arXiv2021

Quasilinear tropical compactifications

The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed subvarieties of tori. We introduce a broader class of tropical compactifications, which we call quasilinear (tropical) compactifications, and which continue to satisfy the desirable properties of compactifications of complements of hyperplane arrangements. In particular, we show any quasilinear compactification is sch\"on, and its intersection theory is described entirely by the intersection theory of the corresponding tropical fan. As applications, we prove the quasilinearity of the moduli spaces of 6 lines in $\mathbb{P}^2$ and marked cubic surfaces, obtaining results on the geometry of the stable pair compactifications of these spaces.

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arXiv2020

Intersection theory of the stable pair compactification of the moduli space of six lines in the plane

We describe sequences of blowups of $\overline{M}_{0,5} \times \overline{M}_{0,5}$ and $\mathbf{P}^2 \times \mathbf{P}^2$ yielding a small resolution of the stable pair compactification $\overline{M}(3,6)$ of the moduli space $M(3,6)$ of six lines in $\mathbf{P}^2$. These blowup sequences can be viewed, respectively, as generalizations of Keel's and Kapranov's constructions of $\overline{M}_{0,n}$. We use these blowup sequences to describe the intersection theory of $\overline{M}(3,6)$. In particular, we show that the Chow ring of any small resolution of $\overline{M}(3,6)$ has a presentation analogous to Keel's presentation of $A^*(\overline{M}_{0,n})$, and the Chow ring of $\overline{M}(3,6)$ is an explicit subring of the Chow ring of one of these small resolutions. We also introduce higher-dimensional versions of the $ψ$-classes on $\overline{M}_{0,n}$, and describe their intersections on $\overline{M}(3,6)$. Finally, we use our results to obtain an independent proof of Luxton's result that $\overline{M}(3,6)$ is the log canonical compactification of $M(3,6)$.

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arXiv2016

Classically Integral Quadratic Forms Excepting at Most Two Values

Let $S \subseteq \mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \emptyset$, this was solved (for classically integral forms) by the $15$-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the $290$-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent $S=\{m\}$; of his $88$ candidates, he could provide complete justifications for all but one. In the same spirit, we ask, "for which $S = \{m, n\}$ does there exist a quadratic form excepting only the elements of $S$?" Extending the techniques of Bhargava and Hanke, we answer this question for quaternary forms. In the process, we prove what Halmos could not; namely, that $x^2+2y^2+7z^2+13w^2$ represents all positive integers except $5$. We develop new strategies to handle forms of higher dimensions, yielding an enumeration of and proofs for the $73$ possible pairs that a classically integral positive definite quadratic form may except.

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