영남 대수기하 워크샵 (2021 가을)

- on the occasion of Sijong Kwak's 60th birthday

Organizers

Invited Speakers

Panel Discussion

Participants

Program

Title & Abstract

이완석 (부경대학교)
Title : A construction of surfaces containing curves of maximal regularity
Abstract : For a nondegenerate projective curve $C \subset \mathbb{P}^r$ of degree d and arithmetic genus g, G. Castelnuovo gave an upper bound $\pi_0(d,r)$ on genus g. In particular, a curve C which attains the maximal genus always lies on a surface of minimal degree. Castelnuovo's theorem has been studied by G. Halphen, G. Fano and recently by Eisenbud-Harris and I. Petrakiev. The main philosophy of their results is that curves of sufficiently high genus should lie on surfaces of some small degree. This is reinterpreted as the relation between the number of quadratic generators of the defining ideals of curves and the degree of surfaces containing those curves. Indeed, if $d \geq 2r+1$ then Castelnuovo's theorem says that $h^0(\mathbb{P}^r, \mathcal{I}_C(2)) \leq {{r-1}\choose{2}}$ and the equality holds if and only if C lies on a surface of minimal degree r-1. To the best of author's knowledge, only next few cases are well understood. In this direction, we study the problem to generalize Castelnuovo's theorem for curves of maximal (Castelnuovo-Mumford) regularity.

최영욱 (영남대학교)
Title : Normal generation of non-special line bundles on multiple coverings
Abstract : In this talk, we discuss very ample non-special line bundles which fail to be normally generated on a simple k-fold covering $\phi : X \to Y$ in terms of line bundles on Y under some numerical constraints. To describe such line bundles, we use the notion minimally FIIQ divisor on a smooth curve X in $\mathbb P^r$. A minimally FIIQ divisor is an effective divisor which fails to impose independent conditions on quadrics in $\mathbb P^r$ and all of whose proper sub-divisors impose independent conditions on quadrics. Moreover, in the cases $g_Y =1$ and $2$, we give a specific classification of such line bundles on X by using the minimally FIIQ property on the base curve Y.

김영훈 (서울대학교)
Title : Algebraic geometry of quantum states
Abstract : Quantum computers are computers based on quantum mechanics in which information is handled by quantum states. As the space of quantum states is a complex projective space, many classical constructions and results in projective geometry are quite useful. In this talk, I will review basics on quantum states and discuss some applications of projective geometry and invariant theory to quantum computing such as entanglement, LOCC and PPT edge states.

김영록 (한국외국어대학교)
Title : 머신러닝을 이용한 과일 가격 예측 알고리즘 연구
Abstract : 한국 과일 가격을 예측하기 위하여 딥러닝 모델들을 제작하였고 모델별로 과일 가격 예측에 대한 정확도를 분석하였다. 과일 가격 예측 모델을 제작하는데 사용한 데이터들은 5가지 과일의 지역별 가격 데이터, 기상 데이터, 종합 주가 지수 데이터들이다. 이 데이터들은 2011년부터 2020년까지 10년 동안의 일자별 시계열 데이터로 Open-API를 통하여 수집하였다. 어떤 데이터가 과일 가격 예측에 유용할지 밝히기 위하여 수집한 데이터들로 몇 가지의 학습 데이터들을 구성하였고 5가지 과일의 가격을 목표값(target)으로 선정하여 각 모델들을 학습하였다. 모델들은 시계열 데이터 예측에 뛰어난 성능을 보이는 LSTM 알고리즘을 사용하여 제작되었다. 우리는 모델들의 성능을 평가하기 위하여 각 모델의 예측값과 실제 과일 가격 사이의 상대 오차율을 지표로 사용하였다. 더불어 자세한 분석을 위하여 2019년과 2020년의 과일 가격을 각각 예측하여 연도별 상대 오차율을 비교하였고 각 연도별로 모델 성능 향상에 유용한 데이터들을 선정하였다. 이를 통하여 효율성이 우수한 모델을 확인한 후 서비스를 제공할 가장 좋은 모델을 제시하였다.

정기룡 (경북대학교)
Title : Double lines in the del-Pezzo 4-fold
Abstract : In this talk, as a completion of the work of Y. Prokhorov, we classify all of the normal bundles of lines in the del-Pezzo 4-fold $Gr(2,5) \cap H_1 \cap H_2$.

김선정 (경상대학교)
Title : On the plane curves attaining the Sziklai bound
Abstract : We consider a curve C in the projective plane $\mathbb{P}^2(\bar{\mathbb{F}_q})$ over the algebraic closure $\bar{\mathbb{F}_q}$ of a finite field $\mathbb{F}_q$, and denote by $C(\mathbb{F}_q)$ the set of $\mathbb{F}_q$-points of C. The curve C needs not be absolutely irreducible, but we suppose it has no $\mathbb{F}_q$-line component. In [Homma-Kim, 2010], we proved that $|C(\mathbb{F}_q)| \le (d-1)q+1$ for every plane curve of degree d with no $\mathbb{F}_q$-linear component, except only one exception. The Sziklai bound means the number $(d-1)q+1$. In this talk, we consider the plane curves attaining the Sziklai bound.

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