Related papers: Contractible Extremal Rays on \overline{M}_{0,n}

We develop a new method for establishing the extremality in the closed cone of effective curves on the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by using a general criterion for…

We establish criteria for when Noether--Lefschetz divisors generate an extremal ray in the cone of pseudoeffective divisors of an orthogonal modular variety. In particular, we exhibit many extremal rays of the cone of pseudoeffective…

We show that certain divisors of Brill-Noether and Gieseker-Petri type span extremal rays of the effective cone in the moduli space of stable genus one curves with $n$ ordered marked points. In particular, they are different from the…

Let X_n := \bar M_{0,n}, the moduli space of n-pointed stable genus zero curves, and let X_{n,m} be the quotient of X_n by the action of the symmetric group S_{n-m} on the last n-m marked points. The cones of effective divisors of X_{n,m},…

We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety of dimension at most 4, every covering and quasi-unsplit family of…

We study a family of semiample divisors on the moduli space $\bar{M}_{0,n}$ that come from the theory of conformal blocks for the Lie algebra $sl_n$ and level 1. The divisors we study are invariant under the action of $S_n$ on…

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.

This is the second of two papers on the birational geometry of $\bar{M}_{g,1}$. We construct rational maps from $\bar{M}_{5,1}$ and $\bar{M}_{6,1}$ to lower-dimensional moduli spaces. As a consequence, we identify geometric divisors that…

We identify a set of initial rational contractions of fiber type on $\overline{M}_{0,6}$. Our proof uses a new algorithm we develop for verifying descriptions of the cone of effective divisors on varieties without elementary rational…

We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet and Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen and Coskun. Results include a…

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families…

We show that the pseudo-effective cone of divisors of $\overline{M}_{0,n}$ is not polyhedral for $n\geq 8$ by constructing an extremal non-polyhedral ray of the dual cone of moving curves via maps on meromorphic strata of differentials…

In this paper we examine the cones of effective cycles on blow ups of projective spaces along smooth rational curves. We determine explicitly the cones of divisors and 1- and 2-dimensional cycles on blow ups of rational normal curves, and…

The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n<=6. It is well known that the class of the closure of the divisor obtained for n=1…

We study cyclic covering morphisms from $\bar{M}_{0,n}$ to moduli spaces of unpointed stable curves of positive genus or compactified moduli spaces of principally polarized abelian varieties. Our main application is a construction of new…

We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to…

We show that the pseudoeffective cone of divisors $\overline{\text{Eff}}^1(\overline{\mathcal{M}}_{g,n})$ for $g\geq 2$ and $n\geq 2$ is not polyhedral by showing that the class of the fibre of the morphism forgetting one point forms an…

We classify extremal divisorial contraction which contracts a divisor to a curve from a smooth fourfold. We prove the exceptional divisor is $\Bbb P^2$bundle or quadric bundle over a smooth curve and the contraction is the blowing up along…

Let $\xi$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$, and let $M(\xi)$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $\xi$. In this paper, we provide an…

This is a continuation of "Rational families of vector bundles on curves, I". Let C be a smooth projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1.…