arXiv subjects

Shin-Yao Jow

Publications and source records attributed to Shin-Yao Jow.

11 recordsLinked to original sources

arXiv2023

Poncelet's theorem for conics in any position and any characteristic

Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of, including Poncelet's original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over the field of complex numbers, or both. Here we show that Poncelet's theorem holds for any two conics C and D in the projective plane over an algebraically closed field k of any characteristic other than two. If C and D are osculating and char(k)>2, our result shows that there always exist infinitely many polygons of char(k) sides that are inscribed in C and circumscribed about D. We also describe the situation in characteristic two in the appendix.

math.AG↗

arXiv2019

Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in $\mathbb{P}^n$

Let $Z$ be a finite set of $s$ points in the projective space $\mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $α(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,\ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $\widehatα(Z)$ of $Z$ is defined by the limit \[ \widehatα(Z)=\lim_{m \to \infty}\frac{α(mZ)}{m}. \] Demailly conjectured that \[ \widehatα(Z)\geq\frac{α(mZ)+n-1}{m+n-1}. \] Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\geq m-1. \] Here we improve their result and show that Demailly's conjecture holds if $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\ge \frac{2\varepsilon}{n-1}(m-1), \] where $0\le \varepsilon<1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$ very general points where $\sqrt[n]{s}\in\mathbb{N}$ (namely $\varepsilon=0$), Demailly's conjecture holds for all $m\in\mathbb{N}$. We also show that Demailly's conjecture holds if $Z$ is very general and \[ s\ge\max\{n+7,2^n\}, \] assuming the Nagata-Iarrobino conjecture $\widehatα(Z)\ge\sqrt[n]{s}$.

math.AG↗

arXiv2019

Asymptotic constructions and invariants of graded linear series

Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\mathbf{K}$. Let $V_\bullet$ be a graded linear series associated to a line bundle $L$ on $X$, that is, a collection $\{V_m\}_{m\in\mathbb{N}}$ of vector subspaces $V_m\subseteq H^0(X,L^{\otimes m})$ such that $V_0=\mathbf{K}$ and $V_k\cdot V_\ell\subseteq V_{k+\ell}$ for all $k,\ell\in\mathbb{N}$. For each $m$ in the semigroup \[ \mathbf{N}(V_\bullet)=\{m\in\mathbb{N}\mid V_m\ne 0\},\] the linear series $V_m$ defines a rational map \[ ϕ_m\colon X\dashrightarrow Y_m\subseteq\mathbb{P}(V_m), \] where $Y_m$ denotes the closure of the image $ϕ_m(X)$. We show that for all sufficiently large $m\in \mathbf{N}(V_\bullet)$, these rational maps $ϕ_m\colon X\dashrightarrow Y_m$ are birationally equivalent, so in particular $Y_m$ are of the same dimension $κ$, and if $κ=n$ then $ϕ_m\colon X\dashrightarrow Y_m$ are generically finite of the same degree. If $\mathbf{N}(V_\bullet)\ne\{0\}$, we show that the limit \[ \operatorname{vol}_κ(V_\bullet)=\lim_{m\in \mathbf{N}(V_\bullet)}\frac{\dim_\mathbf{K} V_m}{m^κ/κ!}\] exists, and $0<\operatorname{vol}_κ(V_\bullet)<\infty$. Moreover, if $Z\subseteq X$ is a general closed subvariety of dimension $κ$, then the limit \[ (V_\bullet^κ\cdot Z)_\text{mov}=\lim_{m\in \mathbf{N}(V_\bullet)}\frac{\#\bigl((D_{m,1}\cap\cdots\cap D_{m,κ}\cap Z)\setminus \operatorname{Bs}(V_m)\bigr)}{m^κ}\] exists, where $D_{m,1},\ldots,D_{m,κ}\in |V_m|$ are general divisors, and \[ (V_\bullet^κ\cdot Z)_\text{mov}=\operatorname{deg}\bigl(ϕ_m|_Z\colon Z\dashrightarrow ϕ_m(Z)\bigr)\operatorname{vol}_κ(V_\bullet) \] for all sufficiently large $m\in\mathbf{N}(V_\bullet)$.

math.AG↗

arXiv2017

On the principally polarized abelian varieties that contain m-minimal curves

In this paper, we study principally polarized abelian varieties $X$ of dimension $g$ that contain a curve $ν:C\to X$ such that the class of $C$ is $m$ times the minimal class. Welters introduced the formalism of stable pairs to handle this problem in the case $m=2$. We generalize the results of Welters and construct families of principally polarized abelian varieties for any $m$ and compute the dimension of the locus of these abelian varieties.

math.AG↗

arXiv2016

Fano varieties with finitely generated semigroups in the Okounkov body construction

The Okounkov body is a construction which, to an effective divisor D on an n-dimensional algebraic variety X, associates a convex body in the n-dimensional Euclidean space R^n. It may be seen as a generalization of the moment polytope of an ample divisor on a toric variety, and it encodes rich numerical information about the divisor D. When constructing the Okounkov body, an intermediate product is a lattice semigroup, which we will call the Okounkov semigroup. Recently it was discovered that finite generation of the Okounkov semigroup has interesting geometric implication for X regarding toric degenerations and integrable systems, however the finite generation condition is difficult to establish except for some special varieties X. In this article, we show that smooth projective del Pezzo varieties have finitely generated Okounkov semigroups, providing the first family of nontrivial higher dimensional examples that are not coming from representation theory. Our result also gives a partial answer to a question of Anderson, Kuronya, and Lozovanu.

math.AG↗

arXiv2011

The effective cone of the space of parametrized rational curves in a Grassmannian

We determine the effective cone of the Quot scheme parametrizing all rank r, degree d quotient sheaves of the trivial bundle of rank n on P^1. More specifically, we explicitly construct two effective divisors which span the effective cone, and we also express their classes in the Picard group in terms of a known basis.

math.AG↗

arXiv2011

Cohomology of toric line bundles via simplicial Alexander duality

We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T. Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on "Serre duality for Betti numbers" which was raised but unresolved in arXiv:1006.2392.

math.AG↗

arXiv2011

Multigraded Fujita Approximation

The original Fujita approximation theorem states that the volume of a big divisor $D$ on a projective variety $X$ can always be approximated arbitrarily closely by the self-intersection number of an ample divisor on a birational modification of $X$. One can also formulate it in terms of graded linear series as follows: let $W_{\bullet} = \{W_k \}$ be the complete graded linear series associated to a big divisor $D$: \[ W_k = H^0\big(X,\mathcal{O}_X(kD)\big). \] For each fixed positive integer $p$, define $W^{(p)}_{\bullet}$ to be the graded linear subseries of $W_{\bullet}$ generated by $W_p$: \[ W^{(p)}_{m}={cases} 0, &\text{if $p\nmid m$;} \mathrm{Image} \big(S^k W_p \rightarrow W_{kp} \big), &\text{if $m=kp$.} {cases} \] Then the volume of $W^{(p)}_{\bullet}$ approaches the volume of $W_{\bullet}$ as $p\to\infty$. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.

math.AG↗

arXiv2010

A Lefschetz hyperplane theorem for Mori dream spaces

Let X be a smooth Mori dream space of dimension at least 4. We show that, if X satisfies a suitable GIT condition which we call "small unstable locus", then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Neron-Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.

math.AG↗

arXiv2009

Okounkov bodies and restricted volumes along very general curves

Given a big divisor $D$ on a normal complex projective variety $X$, we show that the restricted volume of $D$ along a very general complete-intersection curve $C\subset X$ can be read off from the Okounkov body of $D$ with respect to an admissible flag containing $C$. From this we deduce that if two big divisors $D_1$ and $D_2$ on $X$ have the same Okounkov body with respect to every admissible flag, then $D_1$ and $D_2$ are numerically equivalent.

math.AG↗

arXiv2007

Multiplier ideals of sums via cellular resolutions

Fix nonzero ideal sheaves a_1,...,a_r on a normal Q-Gorenstein complex variety X. Fix any positive real number c, and consider the multiplier ideal J of the sum a_1+...+a_r with weighting coefficient c. We construct an exact sequence resolving J by sheaves over X that are direct sums of multiplier ideals for products a_1^{v_1}...a_r^{v_r} for various real vectors v such that v_1+...+v_r = c. The resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation T of the simplex of nonnegative real vectors summing to c. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on T, of a related monomial ideal. This acyclicity rests on a comparison between the homology of certain homology-manifolds-with-boundary and the homology of the simplicial complexes obtained by deleting collections of boundary faces from them. Our resolution implies the multiplier ideal sum formula J((a_1+...+a_r)^c) = \sum_{|v|=c} J(a_1^{v_1}...a_r^{v_r}), which implicitly follows from Takagi's proof of the two-summand formula (math.AG/0410612). We recover Howald's multiplier ideal formula for monomial ideals (math.AG/0003232) as a special case. Our resolution also yields a new exactness proof for the Skoda complex.

math.AG↗