Birational Morphism Research Articles

On the new compactification of moduli of vector bundles on a surface. II

We construct a new compactification of the moduli scheme of Gieseker-stable vector bundles having fixed Hilbert polynomial on a smooth projective polarized surface defined over the field . Families of locally free sheaves on the surface are completed by locally free sheaves on surfaces that are certain modifications of . The new moduli space has a birational morphism onto the Gieseker-Maruyama moduli space. We consider the case when the Gieseker-Maruyama space is the coarse moduli space.Bibliography: 16 titles.

Stability of bicanonical curves of genus three

We work out the geometric invariant theory of the Hilbert scheme and the Chow variety of bicanonically embedded curves of genus three. We show that a Hilbert semistable bicanonical curve has node, ordinary cusp and tacnode as singularity but does not admit elliptic tails or bridges, and that it is Hilbert stable if, moreover, it does not have a tacnode. We prove that a Chow semistable bicanonical curve allows the same set of singularities but no elliptic tail, and that those with an elliptic bridge or a tacnode are identified in the GIT quotient space. We also give a geometric description of the birational morphism from the Hilbert quotient to the Chow quotient.

On the exceptional locus of the birational projections of a normal surface singularity into a plane

Given a normal surface singularity ( X , Q ) and a birational morphism to a non-singular surface π : X → S , we investigate the local geometry of the exceptional divisor L of π. We prove that the dimension of the tangent space to L at Q equals the number of exceptional components meeting at Q. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterisation of minimal singularities is obtained in these terms.

Singular Kähler-Einstein metrics

We study degenerate complex Monge-Ampere equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ where $\omega$ is a big semi-Kahler form on a compact Kahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$ is a positive measure with density $f\in L^p(X,\omega^n)$, $p>1$. We prove the existence and unicity of continuous $\o$-plurisubharmonic solutions. In case $X$ is projective and $\omega=\psi^*\omega'$, where $\psi:X\to V$ is a proper birational morphism to a normal projective variety, $[\omega']\in NS_{\R} (V)$ is an ample class and $\mu$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kahler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

On a new compactification of the moduli of vector bundles on a surface

A new compactification of the moduli scheme of Gieseker-stable vector bundles with prescribed Hilbert polynomial on a smooth projective polarized surface defined over a field of characteristic zero is constructed. The families of locally free sheaves on the surface are completed by locally free sheaves on surfaces that are certain modifications of . The new moduli space has a birational morphism onto the Gieseker-Maruyama moduli space. The case when the Gieseker-Maruyama space is a fine moduli space is considered. Bibliography: 12 titles.

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G ⊂ GL (n, k ), we describe the coherent component Yθ of the moduli space ℳθ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Y θ → A k n / G obtained by variation of Geometric Invariant Theory quotient. As a special case, this gives a new construction of Nakamura's G-Hilbert scheme HilbG that avoids the (typically highly singular) Hilbert scheme of |G|-points in A k n . To conclude, we describe the toric fan of Yθ and hence calculate the quiver representation corresponding to any point ofYθ.

Surfaces with $K^2 \lt 3\chi$ and finite fundamental group

In this paper we continue the study of $\pionealg(S)$ for minimal surfaces of general type $S$ satisfying $K_S^2 <3\chi(S)$. We show that, if $K_S^2= 3\chi(S)-1$ and $|\pionealg(S)|= 8$, then $S$ is a Campedelli surface. In view of the results of \cite{3chi} and \cite{3chi-2}, this implies that the fundamental group of a surface with $K^2<3\chi$ that has no irregular \'etale cover has order at most 9, and if it has order 8 or 9, then $S$ is a Campedelli surface. To obtain this result we establish some classification results for minimal surfaces of general type such that $K^2=3p_g-5$ and such that the canonical map is a birational morphism. We also study rational surfaces with a $\Z_2^3$-action

Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g ⩾ 3 and let M 0 be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M 0 have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan–Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan–Ramanan's, and prove that the Narasimhan–Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.

A Remark on Partial Resolutions of 3-Dimensional Terminal Singularities

Let X be a 3-dimensional terminal singularity of index ≥ 2. We study projective birational morphisms ϕ: Y → X such that the exceptional divisor of ϕ consists of all prime divisors with discrepancies &lt; 1 (resp. ≤ 1) over X.

Gorenstein Resolutions of 3-Dimensional Terminal Singularities

Let X be a 3-dimensional terminal singularity of index ≥ 2. We shall construct projective birational morphisms ƒ: Y → X such that Y has only Gorenstein terminal singularities and that ƒ factors the minimal resolution of a general member of | −KX |. We also study prime divisors of ƒ, especially the discrepancies of these prime divisors.

A Nonlinear Version of Noether's Type Theorem#

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto Φ(C) ⊂ P r . We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H 0(C, L 2) can be written as α2 + β2 + γ2, with α, β, γ ∈ H 0(C, L). If there are no quadrics of rank 3 containing Φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem.

Semistable divisorial contractions

The semistable minimal model program is a special case of the minimal model program concerning 3-folds fibred over a curve and birational morphisms preserving this structure. We classify semistable divisorial contractions which contract the exceptional divisor to a normal point of a fibre. Our results can be applied to describe compact moduli spaces of surfaces.

Birational morphisms of the plane

Let A 2 A^2 be the affine plane over a field K K of characteristic 0 0 . Birational morphisms of A 2 A^2 are mappings A 2 → A 2 A^2 \to A^2 given by polynomial mappings φ \varphi of the polynomial algebra K [ x , y ] K[x,y] such that for the quotient fields, one has K ( φ ( x ) , φ ( y ) ) = K ( x , y ) K(\varphi (x), \varphi (y)) = K(x,y) . Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ x \tau _x given by x → x , y → x y x \to x, ~y \to xy . For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ x \tau _x . This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.

On the Picard modular surface for ℚ(√–2) and related Theta constants

The family of abelian varieties over C which have the endomorphism √–d of type (n, 1) is parameterized on the complex ball Bn. Let G be a modular group for such a family equipped with a certain level structure. The quotient is called the Picard modular variety of level G. There are many studies on Picard modular varieties (for example, see [2]). But concrete models are known only for the cases d = 1, 3 and 163 (see [3], [6], [7] and [9]). In this paper, we study the Picard modular surface for d = 2. In Section 1, we give an explicit modular embedding Φ : Bn → Hn+1 for d ≡ 1, 2 mod 4. This is inspired by the modular embedding given in [7]. In Section 2, we study several congruence subgroups of the Picard modular group G(ℤ) for ℚ(√–2 ) and relations among them. Furthermore we show some special properties of the theta constants restricted on Φ(Bn) with d = 2. In Section 3, we state the main theorem. Namely, (1) We give a birational morphism from \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \overline {{\bf B}_2/ G_2(\mathbb Z)} $\end{document} onto a hypersurface in the weighted projective space P(1, 2, 3, 4). (2) We describe the image as an explicit hypersurface of degree 6 defined over ℤ[√2 ].

Computation of blowing up centers

Given a birational projective morphism of quasi-projective varieties f : Z→X . We want to find the ideal sheaf I over X such that the blowing up of X along I corresponds to f. In this paper we approach the problem from two directions, solving two subcases. First we present a method that determines I when f is the composition of blowing ups along known centers, then by another method, we compute I directly from i(Z)⊂ P X n .

Moduli spaces of weighted pointed stable curves

A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum of the corresponding weights is no greater than one. We construct moduli spaces for these objects using methods of the log minimal model program, and describe the induced birational morphisms between moduli spaces as the weights are varied. In the genus zero case, we explain the connection to Geometric Invariant Theory quotients of points in the projective line, and to compactifications of moduli spaces studied by Kapranov, Keel, and Losev-Manin.

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.

On the birational geometry of toric Fano [formula omitted]-folds

In this Note, we announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the three-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

On birational morphisms between pencils of Del Pezzo surfaces

Let X / S X/S and X ′ / S ′ X’/S’ be two Del Pezzo fibrations of degrees d d , d ′ d’ respectively. Assume that X X and X ′ X’ differ by a flop. Then we prove that d = d ′ d=d’ and give a short list of values of other basic numerical invariants of X X and X ′ X’ .

Global smoothing of Calabi-Yau threefolds

The moduli spaces of Calabi–Yau threefolds are conjectured to be connected by the combination of birational contraction maps and flat deformations. In this context, it is important to calculate dim Def(X) from dim Def(˜X) in terms of certain geometric information of f, when we are given a birational morphism f:˜X→X from a smooth Calabi–Yau threefold ˜X to a singular Calabi–Yau threefold X. A typical case of this problem is a conjecture of Morrison-Seiberg which originally came from physics. In this paper we give a mathematical proof to this conjecture. Moreover, by using output of this conjecture, we prove that certain Calabi–Yau threefolds with nonisolated singularities have flat deformations to smooth Calabi–Yau threefolds. We shall use invariants of singularities closely related to Du Bois's work to calculate dim Def(X) from dim Def(˜X).