Birational Geometry of the Space of Complete Quadrics (original) (raw)

The Birational Geometry of Moduli Spaces

Algebraic Geometry and Number Theory, 2017

In these notes, we discuss the cones of ample and effective divisors on various moduli spaces such as the Kontsevich moduli spaces of stable maps and the moduli space of curves. We describe the stable base locus decomposition of the effective cone in explicit examples and determine the corresponding birational models.

Birational geometry of moduli spaces of configurations of points on the line

arXiv: Algebraic Geometry, 2017

In this paper we study the geometry of GIT configurations of nnn ordered points on mathbbP1\mathbb{P}^1mathbbP1 both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient (mathbbP1)n//PGL(2)(\mathbb{P}^1)^n//PGL(2)(mathbbP1)n//PGL(2), taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of (mathbbP1)n//PGL(2)(\mathbb{P}^1)^n//PGL(2)(mathbbP1)n//PGL(2) in its natural embedding, and its group of automorphisms.

On birational trivial families and Adjoint quadrics

arXiv: Algebraic Geometry, 2019

Let picolonmathcalXtoB\pi\colon \mathcal{X}\to BpicolonmathcalXtoB be a family whose general fiber XbX_bXb gives a (d1,...,da)(d_1,...,d_a)(d1,...,da) polarisation of a general Abelian variety where 1leqdileq21\leq d_i\leq 21leqdileq2, i=1,...,ai=1,...,ai=1,...,a and ageq4a\geq 4ageq4. We show that the fibers are in the same birational class if all the (m,0)(m,0)(m,0) forms on XbX_bXb are liftable to (m,0)(m,0)(m,0) forms on mathcalX\mathcal{X}mathcalX where m=1m=1m=1 and m=a−1m=a-1m=a−1. Actually we show general criteria to find families with fibers in the same birational class, which leads together with a famous theorem of Nori to some interesting applications.

The Birational Geometry of the Hilbert Scheme of Points on Surfaces

Birational Geometry, Rational Curves, and Arithmetic, 2013

In this paper, we study the birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as P 2 , P 1 × P 1 and F 1. We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with K 2 ≥ 2. As a corollary, we show that the Hilbert scheme of points on a Fano surface is a Mori dream space. We then discuss effective divisors on Hilbert schemes of points on surfaces and determine the stable base locus decomposition completely in a number of examples. Finally, we interpret certain birational models as moduli spaces of Bridgeland stable objects. When the surface is P 1 × P 1 or F 1 , we find a precise correspondence between the Mori walls and the Bridgeland walls, extending the results of [ABCH] to these surfaces. Contents 1. Introduction 1 2. The ample cone of the Hilbert scheme 4 3. The effective cone of the Hilbert scheme 10 4. Stable base locus decomposition of the effective cone of X [n] 13 5. Preliminaries on Bridgeland stability and Bridgeland walls 23 6. The correspondence between Bridgeland walls and Mori walls 27 References 32

Birational geometry of terminal quartic 3-folds

2012

1.1. Abstract. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds, a small step inside a larger program of research. A quartic 3-fold is an example of a Fano variety, that is, a variety X with ample anticanonical sheaf OX(−KX).

The usual Castelnuovo's argument and special subhomaloidal systems of quadric

Dedicated to Silvio Greco in occasion of his 60-th birthday. We prove that a linearly normal n-dimensional variety X n ⊂ P r , regular if n ≥ 2, and of degree d ≤ 2(r − n) is the scheme-theoretic intersection of the quadrics through it by using the usual Castelnuovos argument that 2r + 1 points in P r in general linear position impose independent conditions to quadric hypersurfaces; if d ≤ 2(r − n) − 1, we apply the same argument to show that H 0 (I X (2)) gives a special subhomaloidal system whose base locus is clearly X. If moreover Sec(X) ⊂ P r , then the linear system is homaloidal. We apply these results to show that some varieties X n ⊂ P 2n+1 have one apparent double point and we also study the relations between the linear system of quadrics de�ning a variety and the locus of secant lines passing through a general point of the ambient space.

SOME MODULI OF n-POINTED FANO FOURFOLDS

The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the n-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.

Some comments on projective quadrics subordinate to

2010

We study in some detail the structure of the projective quadric Q' obtained by taking the quotient of the isotropic cone in a standard pseudo-Hermitian space H_{p,q} with respect to the positive real numbers R^+ and, further, by taking the quotient Q'/U(1). The case of signature (1,1) serves as an illustration. Q'/U(1) is studied as a compactification of RxH_{p-1,q-1}.