Shigeru Mukai, Hirokazu Nasu
We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled 3-folds and the Hom scheme from a general curve of genus five to a general cubic 3-fold.
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We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the ``algebraic'' deformation functor of an ordinary curve X over a field of positive charateristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X -> G\X. The second one is the ``analytic'' deformation functor of a fixed embedding of a finitely generated discrete group N in PGL(2,K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre represe…
We study the deformations of a curve $C$ on an Enriques-Fano $3$-fold $X \subset \mathbb P^n$, assuming that $C$ is contained in a smooth hyperplane section $S \subset X$, that is a smooth Enriques surface in $X$. We give a sufficient condition for $C$ to be (un)obstructed in $X$, in terms of half pencils and $(-2)$-curves on $S$. Let $\operatorname{Hilb}^{sc} X$ denote the Hilbert scheme of smooth connected curves in $X$. By using the Hilbert-flag scheme of $X$, we also compute the dimension of $\operatorname{Hilb}^{sc} X$ at $[C]$ and give a sufficient condition for $\operatorname{Hilb}^{sc}…
We prove that for every smooth prime Fano $3$-fold $V$, the Hilbert scheme $\operatorname{Hilb}^{sc} V$ of smooth connected curves on $V$ contains a generically non-reduced irreducible component of Mumford type. We also study the deformations of degenerate curves $C$ in $V$, i.e., curves $C$ contained in a smooth anti-canonical member $S \in |-K_V|$ of $V$. We give a sufficient condition for $C$ to be stably degenerate, i.e., every small (and global) deformation of $C$ in $V$ is contained in a deformation of $S$ in $V$. As a result, by using the Hilbert-flag scheme of $V$, we determine the dim…
Following Bloch-Esnault-Kerz and Green-Griffiths' recent works on deformation of algebraic cycle classes, we use Chern character from K-theory to negative cyclic homology to show how to eliminate obstructions to deforming cycles.
We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of $C$ in $V$ to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, we give a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of $(-2)$-curves and elliptic curves on $S$. Applying this result, we prove that the Hilbert scheme $\operatorname{Hilb}^{sc} V_4$ of smooth con…
The (Co)homology of the Deligne-Mumford Moduli Spaces of Marked Rational Curves
本文整理了实带标记有理曲线的Deligne-Mumford模空间（上）同调的关键结果与未解决问题，涵盖拓扑与代数两个方向，并尝试用类operad语言刻画其同调结构。
We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals I, in a polynomial ring A, in terms of number of variables and the degrees of generators, when the dimension of A/I is at most two. This bound improves the one obtained by Caviglia and Sbarra in this case. In the continuation of the examples constructed by Clare D'Cruz and the first author, we use families of monomial curves to construct homogeneous ideals showing that these bounds are quite sharp.
In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.
On the Castelnuovo-Mumford regularity of curve arrangements
该研究探讨复射影平面一般曲线构形的雅可比代数及分次导子模的Castelnuovo-Mumford正则性，提出无需拟齐次性假设的加-删型结果，拓展了相关经典结论的适用范围。
We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.
Obstructions to deforming space curves lying on a del Pezzo surface
本文研究了位于度为4的光滑 del Pezzo 曲面上的空间曲线的变形，给出了曲线无障碍或有障碍的充分条件，并构造了无穷多个 Hilbert 概形的非约化不可约分支。
One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only im…
Blowdowns of the Deligne-Mumford Spaces of Real Rational Curves
本文构造了实有理曲线Deligne-Mumford模空间的一列光滑商爆破，给出Keel爆破构造的实类比，并显式描述了中间空间及三类爆破，为后续完全确定实模空间的有理（上）同调环奠定基础。
We give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R).The study has connections with spectral geometry and combinatorics.
We study a topological obstruction of a very stringy nature concerned with deforming the target space of an $N=2$ non-linear \sm. This target space has a singularity which may be smoothed away according to the conventional rules of geometry but when one studies the associated conformal field theory one sees that such a deformation is not possible without a discontinuous change in some of the correlation functions. This obstruction appears to come from torsion in the homology of the target space (which is seen by deforming the theory by an irrelevant operator). We discuss the link between this …
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