Related papers: Rationality of some tetragonal loci

We illustrate the use of the computer algebra system Macaulay2 for simplifications of classical unirationality proofs. We explicitly treat the moduli spaces of curves of genus g=10, 12 and 14.

We prove rationality of the moduli variety of curves of genus 3.

The new result is the unirationality of the moduli space of curves of genus 14. The method applies to lower genus.

The moduli space $\mathcal{S}_{g, 2n}$ parametrizes pointed curves with spin structure. We prove that $\mathcal{S}_{g, 2}$, $\mathcal{S}_{g, 4}$ and $\mathcal{S}_{g, 6}$ are uniruled for particular values of $g$.

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.…

Let $\L_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $\L_g^G$ for $G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4$, or $SL_2(3)$. We show that for $G \iso…

Let C be a smooth projective curve of genus at least 2 over a field k. Given a line bundle L on C, we consider the moduli stack of rank 2n vector bundles E on C endowed with a nowhere degenerate symplectic form $b: E \otimes E \to L$ up to…

By the technique of 3-fold Mori theory, we prove that the moduli space whose general point parameterizes a couple of a smooth curve of genus 4 and a halfcanonical divisor with vanishing global section is rational.

In this short note, we show that any rational curve passing through the generic point in a moduli space of stable bundles with rank $r$ and fixed determinant on a smooth projective curve of genus $g\ge 4$ has degree (with respect to the…

A lot is known about the moduli space of parabolic bundles over curves of genus $g\geq 2$, but the lower genus cases are notably different. The goal of this article is to study the geometry of the moduli space of semistable parabolic…

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian…

We describe the birational geometry of the moduli space S_g^{-} of odd spin curves (theta-characteristics) for all genera g. The odd spin moduli space is a uniruled variety for g<12, and of general type for g at least 12. Furthermore, for…

In this short note we prove the unirationality of Hurwitz spaces of 6-gonal curves of genus $g$ with $5\leq g\leq 28$ or $g=30,31,33,35,36,40,45$. Key ingredient is a liaison construction in $\PP^1 \times \PP^2$. By semicontinuity, the…

Tikhomirov (2009) proved the irreducibility of the moduli space of mathematical instantons on the projective 3-space for all odd charges. The irreducibility for charges between 1 and 5 was known before. In the present paper, the rationality…

We compute the effective cone for the moduli space of stable rational curves with at most six marked points.

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to…

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 $\mathcal{I}_{g,n}$,…

The moduli space M(r,d) of stable, rank r, degree d vector bundles on a smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A, where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The birational…

We show that the coarse moduli space $\cR_5$ of \'etale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational.

We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.