Samir Canning
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View article: FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$
FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$ Open
For fixed genus g and varying finite marking set A, the gluing and forgetful maps give the spaces of holomorphic forms on the moduli space of stable A-marked curves of genus g has the structure of an FA-module, i.e., a functor from the cat…
View article: Tautological and non-tautological cycles on the moduli space of Abelian varieties
Tautological and non-tautological cycles on the moduli space of Abelian varieties Open
The tautological Chow ring of the moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles cla…
View article: Holomorphic forms and non-tautological cycles on moduli spaces of curves
Holomorphic forms and non-tautological cycles on moduli spaces of curves Open
We prove, for infinitely many values of g and n , the existence of non-tautological algebraic cohomology classes on the moduli space $$\mathcal {M}_{g,n}$$ of smooth, genus- g , n -pointed curves. In particular, when $$n=0$$ , …
View article: The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves
The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves Open
We study the appearances of $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of $\m…
View article: On the Chow and cohomology rings of moduli spaces of stable curves
On the Chow and cohomology rings of moduli spaces of stable curves Open
In this paper, we ask: For which (g, n) is the rational Chow or cohomology ring of \bar{\mathcal{M}}_{g,n} generated by tautological classes? This question has been fully answered in genus 0 by Keel (the Chow and cohomology rings are tauto…
View article: Moduli spaces of curves with polynomial point counts
Moduli spaces of curves with polynomial point counts Open
We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the modu…
View article: Tautological and non-tautological cycles on the moduli space of abelian varieties
Tautological and non-tautological cycles on the moduli space of abelian varieties Open
The tautological Chow ring of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes fro…
View article: The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein
The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein Open
We prove that the tautological rings $\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})$ and $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq 2$ and $2g+n\geq 24$, extending results of Petersen and Tommasi in genus $2$…
View article: Holomorphic forms and non-tautological cycles on moduli spaces of curves
Holomorphic forms and non-tautological cycles on moduli spaces of curves Open
We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results sh…
View article: Tautological projection for cycles on the moduli space of abelian varieties
Tautological projection for cycles on the moduli space of abelian varieties Open
We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties $\mathcal{A}_g$: every cycle class decomposes canonically as a sum of a tautological and a non-tautolog…
Extensions of tautological rings and motivic structures in the cohomology of ${\overline {\mathcal {M}}}_{g,n}$ Open
We study collections of subrings of $H^*({\overline {\mathcal {M}}}_{g,n})$ that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension a…
View article: The Chow ring of the moduli space of degree $2$ quasi-polarized K3 surfaces
The Chow ring of the moduli space of degree $2$ quasi-polarized K3 surfaces Open
We study the Chow ring with rational coefficients of the moduli space $\mathcal F_{2}$ of quasi-polarized $K3$ surfaces of degree $2$. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group…
View article: Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$
Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$ Open
We study collections of subrings of $H^*(\overline{\mathcal{M}}_{g,n})$ that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and c…
View article: The Chow rings of moduli spaces of elliptic surfaces over ${\mathbb P}^1$
The Chow rings of moduli spaces of elliptic surfaces over ${\mathbb P}^1$ Open
Let E_N denote the coarse moduli space of smooth elliptic surfaces over P¹ with fundamental invariant N. We compute the Chow ring A∗(E_N) for N ⩾ 2. For each N ⩾ 2, A∗(E_N) is Gorenstein with socle in codimension 16, which is surprising in…
The eleventh cohomology group of Open
We prove that the rational cohomology group $H^{11}(\overline {\mathcal {M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$ . We show furthermore that $H^k(\overline {\mathcal {M}}_{g,n})$ is pure Hodge–Tate for all even $k \leq 12$ and d…
View article: The bielliptic locus in genus 11
The bielliptic locus in genus 11 Open
The Chow ring of $\mathcal{M}_g$ is known to be generated by tautological classes for $g \leq 9$. Meanwhile, the first example of a non-tautological class on $\mathcal{M}_{g}$ is the fundamental class of the bielliptic locus in $\mathcal{M…
View article: The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$
The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$ Open
We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# …
View article: On the Chow and cohomology rings of moduli spaces of stable curves
On the Chow and cohomology rings of moduli spaces of stable curves Open
In this paper, we ask: for which $(g, n)$ is the rational Chow or cohomology ring of $\overline{\mathcal{M}}_{g,n}$ generated by tautological classes? This question has been fully answered in genus $0$ by Keel (the Chow and cohomology ring…
View article: The rational Chow rings of moduli spaces of hyperelliptic curves with marked points
The rational Chow rings of moduli spaces of hyperelliptic curves with marked points Open
We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, pa…
View article: The integral Chow rings of moduli of Weierstrass fibrations
The integral Chow rings of moduli of Weierstrass fibrations Open
We compute the Chow rings with integral coefficients of moduli stacks of minimal Weierstrass fibrations over the projective line. For each integer $N\geq 1$, there is a moduli stack $\mathcal{W}^{\mathrm{min}}_N$ parametrizing minimal Weie…
View article: The integral Picard groups of low-degree Hurwitz spaces
The integral Picard groups of low-degree Hurwitz spaces Open
We compute the Picard groups with integral coefficients of the Hurwitz stacks parametrizing degree $4$ and $5$ covers of $\mathbb{P}^1$. As a consequence, we also determine the integral Picard groups of the Hurwitz stacks parametrizing sim…
View article: The Chow rings of moduli spaces of elliptic surfaces over $\mathbb{P}^1$
The Chow rings of moduli spaces of elliptic surfaces over $\mathbb{P}^1$ Open
Let $E_N$ denote the coarse moduli space of smooth elliptic surfaces over $\mathbb{P}^1$ with fundamental invariant $N$. We compute the Chow ring $A^*(E_N)$ for $N\geq 2$. For each $N\geq 2$, $A^*(E_N)$ is Gorenstein with socle in codimens…
View article: Chow rings of low-degree Hurwitz spaces
Chow rings of low-degree Hurwitz spaces Open
While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space $\mathcal{H}_{k, g}$ parametrizing smooth degree…
View article: The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
The Chow rings of the moduli spaces of curves of genus 7, 8, and 9 Open
The rational Chow ring of the moduli space $\mathcal{M}_g$ of curves of genus $g$ is known for $g \leq 6$. Here, we determine the rational Chow rings of $\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ by showing they are tautological. …
View article: Intersection theory on low-degree Hurwitz spaces
Intersection theory on low-degree Hurwitz spaces Open
While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space $\mathscr{H}_{k, g}$ parametrizing smooth degree…
View article: Tautological classes on low-degree Hurwitz spaces
Tautological classes on low-degree Hurwitz spaces Open
Let $\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of $\mathbb{P}^1$. We define the tautological ring of $\mathcal{H}_{k,g}$ and we show that all Chow classes, except possibly those supported on the loc…
View article: On a Conjecture on the Variety of Lines on a Fano Complete Intersection
On a Conjecture on the Variety of Lines on a Fano Complete Intersection Open
The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove …