Program & Reports

December 4th (Wed)

10:00-10:50  Kaji, Hajime (Waseda University)  楫　元 （早稲田大学）Report

Title：Power of ideals

Abstract：多項式環のイデアルの冪の生成元について考察する．

11:10-12:00  Nasu, Hirokazu (Tokai University) 那須　弘和（東海大学）Report

Title: Stably degenerated and obstructed curves lying on a del Pezzo surface

Abstract: S.Mukai and I have computed obstructions to deforming curves on a threefold, assuming that the curve is contained in a smooth surface with (-1)-curves, e.g., a smooth cubic surface in P^3. Toward the generalization of our result to curves on a higer dimensional algebraic vareity, we consider the problem of determining the deformations of space curves sitting on anticanonically embedded del Pezzo surface of degree > 3. We generalize a conjecture due to J.O.Kleppe and Ph.Ellia and discuss the degree 4 case, i.e., curves on a smooth complete intersection of two quadrics in P^4. We give a sufficient condition for such a curve to be stably degenerate, i.e., every its small deformation is contained in a degree 4 del Pezzo surface.

14:00-14:50 Choi, Youngook (Yeungnam Univ)

Title:Brill-Noether loci of rank-two vector bundles on general $¥nu$-gonal curves

Abstract:In this talk we discuss the Brill-Noether loci for rank-two, stable vector bundles with specialities $2$, focusing on the irreducible components of these loci within  the range $2g-2¥leq d¥leq 4g-4$. We provide a complete classification of the irreducible components, and if time permits, we will demonstrate the existence of irreducible components with specialities $3$, featuring different types (e.g., regular and superabundant, or regular components with different general points). This is joint work with Professor Flaminio Flamini at Universita' degli Studi di Roma Tor Vergata and Professor Seonja Kim at Chungwoon university.

15:10-16:00 Hara, Nobuo(Tokyo University of Agriculture and Technology)原　伸生（東京農工大学）Report

Title: Looking out for Frobenius summands on a quintic del Pezzo surface II

Abstract: This is a continuation of my talk in Aso last year. I will explain an approach to describe the direct sum decomposition of the Frobenius direct images $F^e_*(L^n)$ on a smooth quintic del Pezzo surface $X$ in characteristic $p>0$ with respect to the anti-canonical polarization $L=-K_X$. A few years ago, Mallory has shown that there are only finitely many Frobenius summands (i.e., indecomposable vector bundles appearing in the decomposition) up to twists by $L$. But it is not clear from Mallory's method what kind of bundles appear as Frobenius summands. We are aiming at the identification of all Frobenius summands and detailed description thereof.

16:30-17:20 Takagi, Shunsuke (The University of Tokyo) 高木　俊輔 （東京大学）

Title: Threefolds of globally F-regular type with nef anti-canonical divisor

Abstract: Globally F-regular varieties form a special class of Frobenius split varieties, with examples including toric and Schubert varieties in positive characteristic. Schwede and Smith conjectured a correspondence between globally F-regular varieties and complex Fano-type varieties via reduction to positive characteristic. In this talk, I will explain recent developments related to their conjecture, based on joint work with Paolo Cascini and Tatsuro Kawakami.

December 5th (Thu)

10:00-10:50 Kurano, Kazuhiko (Meiji University) 蔵野　和彦（明治大学）Report

Title: Finite generation of the Cox ring of the blow-up of a weighted projective space

Abstract: In the last 40 years, many researchers have studied the symbolic Rees rings of space monomial primes. For example, in 1994, Goto, Nishida, and Watanabe found the first example that is not finitely generated. This problem is closely related to the Nagata conjecture, and researches on them are still ongoing. In this talk, we will introduce some recent results.

11:10-12:00 Hashimoto, Mitsuyasu (Osaka Metropolitan University) 橋本　光靖 （大阪公立大学）Report

Title: Quasi-Gorenstein property and a-invariants of the ring of invariants

Abstract: 1973年の論文で渡辺敬一先生は、線形簡約な (位数が基礎体の標数で割れない) GL(V) の有限部分群 G について、不変式環　k[V]^G がGorenstein で G が擬鏡映を持たない必要十分条件は G ⊂ SL(V) であることを示されました。その後、この結果はモジュラーな場合 (つまり、G が線形簡約でない場合) や G が被約とは限らない場合など、適切な条件の修正のもとに一般化されてきました。ここでは、G の作用が小さい(余次元２以上を除いて作用が free) 場合に、G の単位連結成分が線形簡約ならば不変式環 k[V]^G が quasi-Gorenstain であることと G ⊂ SL(V) が同値であることを示して、Braun, Fleischmann-Woodcock の結果と Liedtke-Yasuda の結果を同時に一般化するとともに、Gの単位連結成分が線形簡約の条件を外すと反例があることを示します。また、有限群の作用についての Goel-Jeffries-Singh の有限群に関するa不変量に関する結果の一部を有限群スキームに拡張します。

14:00-14:50 Fukuma, Yoshiaki (Kochi University) 福間　慶明 （高知大学）Report

Title: Properties on the dimension of global sections of adjoint bundles for polarized manifolds

Abstract: Let $X$ be a smooth projective variety defined over the field of complex numbers and let $L$ be an ample line bundle on $X$. Recently, the positivity of dimension of $H^{0}(K_{X}+mL)$ for some positive integer $m$ has been discussed. In this talk, I will explain some properties on the dimension of $H^{0}(K_{X}+mL)$.

15:10-16:00 Park, Jinhyung (KAIST)    Report

Title: Positivity of double point divisors

Abstract: The non-isomorphic locus of a general projection from an embedded smooth projective variety to a hypersurface moves in a linear system of an effective divisor which we call the double point divisor. David Mumford proved that the double point divisor from outer projection is always base point free, and Bo Ilic proved that it is ample except when a given variety is a Roth variety. We show that it is very ample except in Roth cases. This answers a question of Bo Ilic. Unlike the case of outer projection, the double point divisor from inner projection may not be base point free nor ample. Atsushi Noma proved that it is semiample except when a given variety is neither a Roth variety, a scroll over a curve, nor the second Veronese surface. We investigate when it is base point free or big. This is joint work with Yonghwa Cho. 

16:30-17:20 Ogata, Shoetsu (Tohoku University) 尾形　庄悦 （東北大学）Report

Title: On Ideals defining Toric Varieties

December 6th (Fri)

10:00-10:50 Fujisawa, Taro (Tokyo Denki University) 藤澤　太郎 （東京電機大学）Report

Title: Semipositivity theorems for variations of mixed Hodge structure 

Abstract: I will discuss about generalization of Fujita--Zucker--Kawamata semipositivity theorem. Prof. Brunebarbe proposed two generalizations, one for polarized variation of (pure) Hodge structures and the other for gradedly polarized admissible variation of mixed Hodge structures. However, there exists a counter-example for the latter. I propose another generalization for gradedly polarized admissible variation of mixed Hodge structures, by modifying Brunebarbe's formulation slightly.

11:10-12:00 Park, Euisung (Korea University)

Title : On rank 3 quadratic equations of Veronese embeddings

Abstract : Many classical constructions in projective geometry, such as rational normal scrolls, Veronese varieties and Segre varieties of more than two projective spaces, are determinantally presented in the sense that their homogeneous ideals are generated by 2-minors of a 1-generic matrix of linear forms. Furthermore, any Segre–Veronese variety is defined ideal-theoretically by 2-minors of several linear determinantal presentations. Therefore the homogeneous ideals of all these varieties are generated by quadratic polynomials of rank 3 and 4. Recently, it has been proven that the ideals of many projective varieties can be generated only by quadratic polynomials of rank 3. In this talk, I aim to introduce such projective varieties, focusing specifically on the Veronese surface.

14:00-14:50 Abe, Takuro (Rikkyo Univ) 阿部　拓郎 （立教大学 ）Report

Title : Castelnouvo-Mumford regularity and Solomon-Terao polynomials of hyperplane arrangements

Abstract : A hyperplane arrangement is a finite set of linear hyperplanes in a vector space, whose origin is the reflecting hyperplane arrangements of Weyl groups. A central topic on algebra of hyperplane arrangements is so called the logarithmic vector fields, which is a generalization of the (co)invariant ring of the Weyl group. In particular, as a generalized coinvariant algebra, the Solomon-Terao algebra and polynomial of hyperplane arrangements was defined and proved that it is isomorphic to the cohomology ring of the regular nilpotent Hessenberg varieties. Thus they are becoming more important in the research of hyperplane arrangements.

In this talk, we investigate the Solomon-Terao polynomial, which is the Hilbert polynomial of the Solomon-Terao algebra in general. We determine the degree of that polynomial by using and investigating the Castenlouvo-Mumford regularity of the logarithmic vector fields based on the results by M. Saito and Bath.

15:10-16:00 Kwak, Sijong (KAIST)

Title: Introduction to higher secant varieties of minimal degree 

and del Pezzo secant varieties

Abstract: There are two basic objects in projective algebraic geometry : 

one is a variety of minimal degree and the other is a del Pezzo variety. In this talk, I'd like to introduce higher secant varieties of minimal degree and del Pezzo higher secant varieties to non-expert with modest backgrounds. After Ciliberto and Russo(2006), classification and characterization of such varieties have been focused recently. I will also introduce many interesting examples explaining main results.

16:20-17:10 Miyazaki, Chikashi (Kumamoto University) 宮崎　誓 （熊本大学）Report

Title: Castelnuovo-Mumford regularity and syzygy theoretic approach to vector bundles

Abstract: I begin with the Castelnuovo-Mumford regularity, which has been mostly occupied in my research, especially describing regularity basics and a ring-theoretic approach to upper bounds on the regularity. After having some comments on what I should have tried, I am going to explain the splitting of vector bundles, characterizing the null-correlation bundles from the  view point of quasi-Buchsbaum bundles.

19:00- Conference reception at Korantei shimotori-ten (open 18:45-)   Congratulatory words by Prof. Kaji

December 7th (Sat)

10:00-10:50 Terai, Naoki (Okayama University) 寺井　直樹 （岡山大学）Report

Title: Depth of symbolic powers of edge ideals of graphs

Abstract: We talk about some  results on the depth descending problem on symbolic powers of edge ideals of simple graphs, including the case of bipartite graphs and graphs with a leaf. This  is based on a joint work with K.Kimura and S. Yassemi.

11:10-12:00 Yanagawa, Kohji (Kansai University) 柳川　浩二 （関西大学）Report

Title: 　Arithmetic and combinatorics on q-deformed rationals 

Abstract: Recently, Morier-Genoud and Ovsienko introduced a q-analog R(q)/S(q) of a rational number (irreducible fraction) r/s > 0. Here R(q) and S(q)  are monic polynomials in ZZ_{>0}[q] satisfying R(1)=r and S(1)=s. While this subject enjoys many applications (e.g., Jones polynomials of rational knots, some 2 Clabi-Yau categories ....), also has its own combinatorial interest. In this talk, we show that, for example, R(q) is symmetric iff s divides r^2-1. If 3 (resp. 4) divides r, then R((-1+sqrt{-3})/2)=0 (resp. R(sqrt{-1})=0). Similar are true for S(q) . 

This is a joint work with Kogiso, Miyamoto, Ren and  Wakui. 

14:00-14:50 Shimomoto, Kazuma (Institute of Science Tokyo) 下元　数馬（東京科学大学）

Title: Lim Cohen-Macaulay sequences and singularities in mixed characteristic

Abstract: This talk will be a report on the recent progresses of constructing lim Cohen-Macaulay sequences over Noetherian local rings of mixed characteristic.  I also discuss the relation of lim Cohen-Macaulay sequences with perfectoid towers. This is a joint work with S. Ishiro.

15:10-16:00 Okada, Takuzo (Kyushu University) 岡田　拓三 （九州大学）

Title: Rationality problem of Johnson-Koll\’{a}r Fano 3-folds

Abstract: In 2001, J. Johnson and J. Koll\’{a}r classified anticanonically embedded quasi-smooth well-formed Fano hypersurfaces in weighted projective 4-spaces into 25 infinite series of deformation families and 4442 sporadic deformation families. In this talk I will report the current status of a joint work (in progress) with Ivan Cheltsov on determining their (ir)rationality.   

16:30-17:30 Free Discussion

※ The reception party will be held at  Friday, December 6, 2024 from 19:00, to celebrate the 65th birthday of Professor Chikashi Miyazaki. If you are interested in attending, please register by Wednesday, November 27.

※１２月６日（金）の夜に，宮崎誓先生の65歳を記念する懇親会を予定しております．参加いただける方は，11月 27日までにwebで登録をお願いいたします．