The ideal of relations for the ring of invariants of n points on the line

Ben Howard; John Millson; Andrew Snowden; Ravi Vakil

Home/Archive/arXiv

arXiv · 0909.3230

The ideal of relations for the ring of invariants of n points on the line

Ben Howard·John Millson·Andrew Snowden·Ravi Vakil

View original source ↗Read source document ↗Export metadata ↓

Abstract

The study of the projective coordinate ring of the (geometric invariant theory) moduli space of n ordered points on P^1 up to automorphisms began with Kempe in 1894, who proved that the ring is generated in degree one in the main (n even, unit weight) case. We describe the relations among the invariants for all possible weights. In the main case, we show that up to the symmetric group symmetry, there is a single equation. For n not 6, it is a simple quadratic binomial relation. (For n=6, it is the classical Segre cubic relation.) For general weights, the ideal of relations is generated by quadratics inherited from the case of 8 points. This paper completes the program set out in [HMSV1].

Explore related subjects

math.AG

Keep this discovery

BibTeX RIS

Ben Howard, John Millson, Andrew Snowden, Ravi Vakil. 2009-09-17. The ideal of relations for the ring of invariants of n points on the line. https://arxiv.org/abs/0909.3230

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

math.AG↗

arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗

arXiv2026

Graded Betti numbers of general curves of large degree

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--S\"{o}derberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

math.AG↗