Lecture Series by Frank-Olaf Schreyer (Saarland University)
Sep 5(Thu), 10(Tue), 12(Thu), 19(Thu), Oct 1(Tue), 15(Tue), 17(Thu), 22(Tue), 24(Thu), 29(Tue) 10:30 - 11:30
Place: Natural Science Building (E6) Room 1410 (Edu4.0)
Title: An experimental and computational approach to algebraic geometry.
Abstract: The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.
YouTube
Lecture 1 (September 5 Thursday)
Lecture 2 (September 10 Tuesday)
Lecture 3 (September 12 Thursday)
Lecture 4 (September 19 Thursday)
Lecture 5 (October 1 Tuesday) Caution: no audio in this recording
Lecture 6 (October 15 Tuesday) [Macaulay2 script]
Lecture 7 (October 17 Thursday)
Lecture 8 (October 22 Tuesday)
Lecture 9 (October 24 Thursday)
Lecture 10 (October 29 Tuesday) [Problem List]
Lecture Series by Lawrence Ein (University of Illinois at Chicago)
Lecture 1: October 24 (Thursday) 14:00 - 15:00,
Title: The Green-Lazarsfeld theorem on projective normality
Abstract: We'll present a proof of the theorem of Green and Lazarsfeld on projective normality using the bend and break trick of Mori (an argument due to Mukai and Sakai).
Lecture 2: October 28 (Monday) 14:30 - 15:30
Title: Kemeny's proof of Voisin's theorem on the syzygies of a generic canonical curve
Abstract: We'll present the beautiful proof of Kemeny on Voisin's theorem using a rank 2 bundle on a K3 surface.
Lecture 3: October 30 (Wednesday) 14:30 - 15:30
Title: Asymptotic syzygies
Abstract: We'll present the results of Ein-Lazarsfeld and Park on asymptotic syzygies of a higher dimensional variety.
Place: Natural Science Building (E6) Room 1401
Seminar (online)
Sep 13(Fri) 10:30 - 11:30
Steven V Sam (University of California, San Diego)
Title: From total positivity to pure free resolutions
Abstract: I will describe recent joint work with Keller VandeBogert on constructing pure free resolutions over quadric hypersurface rings. Along the way I will describe some connections between total positivity and Koszul algebras and some conjectures regarding the homotopy Lie algebra and its "fattened" versions.
Seminar
Sep 20(Fri) 11:30 - 12:30
Place: Natural Science Building (E6) Room 1401
Yeongrak Kim (Pusan National University)
Title : Resonance, Syzygies, and Rank-3 Ulrich bundles on V_5
Abstract : An Ulrich bundle E on an n-dimensional projective variety (X, O(1)) is a vector bundle whose module of twisted global sections is a maximal Cohen-Macaulay module having the maximal number of generators in degree 0. It was once studied by commutative algebraists, but after Eisenbud and Schreyer introduced its geometric viewpoint, many people discovered several important applications in wide areas of mathematics. In this motivating paper, Eisenbud-Schreyer asked a question whether a given projective variety has an Ulrich bundle, and what is the minimal possible rank of an Ulrich bundle if exists. The answer is still widely open for algebraic surfaces and higher dimensional varieties.
Thanks to a number of studies, the answer for the above question is now well-understood for del Pezzo threefolds. In particular, a del Pezzo threefold V_d of (degree d≥3) has an Ulrich bundle of rank r for every r at least 2. The Hartshorne-Serre correspondence translates the existence of rank-3 Ulrich bundle into the existence of an ACM curve C in V_d of genus g=2d+4 and degree 3d+3. In this talk, we first recall a construction of rank-3 Ulrich bundle on a cubic threefold by Geiss and Schreyer, by showing that a "random" curve of given genus and degree lies in a cubic threefold and satisfies the whole conditions we needed. We also discuss how this problem is related to the unirationality of the Hurwitz space H(k, 2g+2k-2) and the moduli of curves M_g. An analogous construction works for d=4, however, for d=5 a general curve of genus 14 and degree 18 does not belong to V_5. We characterize geometric conditions when does such a curve can be embedded into V_5 using the vanishing resonance. This is a joint work with Marian Aprodu.
Seminar
Oct 2(Wed) 16:30 - 17:30
Place: Industrial Engineering & Management Building (E2) Room 1228
Jaewoo Jung (IBS-CCG)
Title: Syzygies of Stanley-Reisner Ideals
Abstract: The study of monomial ideals is central to many areas of commutative algebra and algebraic geometry, with Stanley-Reisner theory providing a crucial bridge between algebraic invariants and combinatorial structures. We explore how the syzygies and Betti diagrams of Stanley-Reisner ideals can be understood through combinatorial operations on simplicial complexes.
In this talk, we focus on the regularity of Stanley-Reisner ideals. We introduce a graph decomposition that bounds the regularity and a decomposition of simplicial complexes with respect to facets. In addition, we introduce secant complexes corresponding to the joins of varieties defined by Stanley-Reisner ideals and investigate the secant variety of minimal degree defined by the Stanley-Reisner ideals. This talk includes multiple collaborative works with G. Blekherman, J. Choe, J. Kim, M. Kim, and Y. Kim.
Seminar
Oct 4(Fri) 10:30 - 11:30
Place: Natural Science Building (E6) Room 1401
Junho Choe (KIAS)
Title: Quadratic equations of tangent varieties and 4-way tensors of linear forms
Abstract: In this talk we present a construction of quadratic equations and their weight one syzygies of tangent varieties using 4-way tensors of linear forms. This is in line with the 2-minor technique for quadratic equations of projective varieties and with the Oeding-Raicu theorem on equations of tangent varieties to Segre-Veronese varieties. We also discuss generalizations of the method if time permits. This is an early stage research.
Seminar
Oct 7(Mon) 16:30 - 17:30
Place: Natural Science Building (E6) Room 1401
Doyoung Choi (KAIST)
Title: Degree of the 3-secant variety
Abstract: We present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles. Additionally, we compute the Hilbert-Samuel multiplicity of 2-secant variety along given variety.
Mini-Workshop 1
Date: Oct 10 (Thu)
Place: Natural Science Building (E6) Room 1401
10:30 - 11:30 Daniele Agostini (University of Tübingen)
Title: Ulrich sheaves on secants of curves
Abstract: We use the geometry of symmetric products of curves to construct rank one symmetric Ulrich sheaves on the (higher) secant varieties. Time permitting, we will also give an application towards an algebraic theory of knots. This is joint work with M. Kummer and J. Park.
14:00 - 15:00 Kangjin Han (DGIST)
Title: On the ideals of $k$-secants of some Veronese embeddings
Abstract: In this talk, we report some results on equations and the ideal of $\sigma_k(v_d(\mathbb{P}^n))$, the $k$-th secant variety of $d$-uple Veronese embedding of a projective space, in case of the $k$-th secant having a relatively small degree. Knowledge on defining equations of higher secant varieties is fundamental in the study of algebraic geometry and in recent years it also has drawn a strong attention in relation to tensor rank problems.
We first recall known results on the equation of a $k$-th secant variety and introduce key notions for this work, which are '$k$-secant variety of minimal degree' and 'del Pezzo $k$-secant variety', due to Ciliberto-Russo and Choe-Kwak, respectively. Next, we focus on the case of $\sigma_4(v_3(\mathbb{P}^3))$ in $\mathbb{P}^{19}$ as explaining our method and considering its consequences. We present more results which can be obtained by the same method. This is a joint work with K. Furukawa (Josai Univ.).
15:30 - 16:30 Michael Kemeny (University of Wisconsin - Madison)
Title: Projections of syzygies
Abstract: I will explain the notion of projection of syzygies, which was originally given by Ehbauer and later was much used by many mathematicians and then give two applications of it. We will firstly explain how it can be used to study the syzygies of canonical curves and in particular explain its application to a conjecture by Schreyer on the ranks of generating linear syzygies for general canonical curves. We will then explain an application of it to the study of linear syzygies of Veronese varieties.
Mini-Workshop 2
Date: Oct 16 (Wed)
Place: Natural Science Building (E6) Room 1410 (Edu4.0)
11:00 - 12:00 Justin Lacini (Princeton University)
Title: Syzygies of adjoint linear series on projective varieties
Abstract: Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In recent joint work with Purnaprajna Bangere we give a positive answer to this question.
14:00 - 15:00 Hyunsuk Moon (KIAS)
Title: Rank 3 generators of Segre-Veronese Embedding
Abstract: Let L be a ample line bundle on a projective scheme X. We say that (X,L) satisfies property QR(k) if the homogeneous ideal can be generated by quadrics of rank less than or equal to k. In the previous paper, we show that the Veronese embedding satisfies property QR(3). Let (X,L) be a Segre-Veronese embedding where X is a product of P^{a_i} with i=1,...,l and L is a very ample lines bundle O_X(d_1,d_2,...,d_l). In the paper [Linear determinantal equations for all projective schemes, SS2011], they prove that (X,L) satisfies QR(4) and it is determinantally presented if at least l-2 entries of d_1,...,d_l are at least 2. in this talk, we prove that (X,L) satisfies Qr(3) if and only if all the entries of d_1,...,d_l are at least 2. For one direction, we compute the radical ideal of 4 by 4 minors of a big matrix with linear forms, and for the other direction, we use the inducution on the sum of entries of (d_1,...,d_l).
15:30 - 16:30 Claudiu Raicu (University of Notre Dame)
Title: Stable sheaf cohomology on flag varieties
Abstract: A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. By embedding smaller flag varieties as Schubert subvarieties in larger ones, one can compare cohomology groups on different spaces and study their eventual asymptotic behavior. In this context I will describe a sharp stabilization result, and discuss some consequences and illustrative examples. Joint work with Keller VandeBogert.
17:00 - 18:00 John Sheridan (Princeton University)
Title: Torelli theorems for certain Steiner bundles on projective space
Abstract: A vector bundle on projective space is called "Steiner" if it can be recognized simply as the cokernel of a map given by a matrix of linear forms. Such maps arise from various geometric setups and one can ask: from the Steiner bundle, can we recover the geometric data used to construct it? In this talk, we will mention an interesting Torelli-type result of Dolgachev and Kapranov from 1993 that serves as an origin of this story, as well as other work that this inspired. We'll then indicate our contribution which amounts to analogous Torelli-type statements for certain tautological bundles on the very ample linear series of a polarized smooth projective variety. This is joint work with R. Lazarsfeld.
Colloquium
Oct 17(Thu) 16:15-17:15
Place: Natural Science Building (E6) Room 1501
Frank-Olaf Schreyer (Saarland University)
Title: Classification of 3-linear varieties of small codimension
Abstract: 2-linear varieties are a rich topic. Sijong Kwak initiated the study of 3-regular varieties. In this talk I report on joined work Haoang Le Truong on the classification of smooth 3-regular varieties of small codimension 3. Some of these varieties are analogously to the 2-regular case determinantal. This first non-determinantal cases occurs in codimension 3. In this talk I report on the classification of varieties with Betti table
$$
\begin{matrix}
& 0 & 1 & 2 & 3\\ \hline
0: & 1 & . & . & .\\
1: & . & . & . & .\\
2: & . & 10 & 15 & 6
\end{matrix}
$$
Our approach consist of studying extension starting from curves. Let $X \subset \mathbb P^n$ be a variety. An e-extension $Y \subset \mathbb P^{n+e}$ of $X$ is a variety, which is not a cone, such that there exists a regular sequence $y_1,\ldots,y_e$ of linear forms for the homogeneous coordinate ring $S_Y$ of $Y$ such that $S_Y/(y_1,\ldots,y_e) = S_X$ is the coordinate ring of $X$. Using a computationally easy deformation theoretic method to compute extensions, we classify the extensions of 3-regular curves in $\mathbb P^4$ to surfaces in $\mathbb P^5$ completely.
Seminar
Oct 18(Fri) 10:30 - 11:30
Place: Natural Science Building (E6) Room 1401
Yonghwa Cho (Gyeongsang National University)
Title: The structure of algebraic surfaces with maximal number of nodes
Abstract: In the 19th century, Kummer extensively studied quartic surfaces in the complex projective 3-space containing 16 nodes(=ordinary double points). One of his notable results states that a quartic surface cannot contain more than 16 nodes. This leads to a classic question: how many nodes may a surface of degree d contain? The answer to this question is known only for a very low degrees, namely, degrees 5 and 6. To find the optimal answer(31) for quintics, Beauville introduced the concept of "even sets of nodes," which turned out to be highly influential in the study of nodal surfaces. Based on the structure theorem of even sets by Casnati and Catanese, we will discuss some structure theorems of nodal quintics and sextics with maximal number of nodes. This talk is based on joint works with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.
Seminar (online)
Oct 23 (Wed) 16:30 - 17:30
Marian Aprodu (University of Bucharest & IMAR)
Title: Resonance and vector bundles
Abstract: Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on works with G. Farkas, Y. Kim, C. Raicu, A. Suciu, and J. Weyman I report on some recent results concerning the geometry of resonance schemes in the vector bundle setup.
Seminar (online)
Oct 30 (Wed) 10:00 - 11:00
Eric Larson (Brown University)
Title: Normal bundles of rational curves in Grassmannians
Abstract: Let $C$ be a general rational curve of degree $d$ in a Grassmannian $G(k, n)$. The natural expectation is that its normal bundle is balanced, i.e., isomorphic to $\bigoplus O(e_i)$ with all $|e_i - e_j| \leq 1$. In this talk, I will describe several counterexamples to this expectation, propose a suitably revised conjecture, and describe recent progress towards this conjecture.
Mini-Workshop 3
Date: Nov 25 (Mon)
Place: Natural Science Building (E6) Room 1410 (Edu4.0)
14:00 - 15:00 Steven Dale Cutkosky (University of Missouri, Columbia)
Title: Asymptotic behavior of Hilbert functions of divisorial filtrations
Abstract: Let (R,m_R) be a d-dimensional, excellent, normal local ring. A divisorial filtration {I_n} is determined by a divisor D on a normal scheme X determined by blowing up an ideal on R, so that I_n are the global sections of nD. Associated to an m_R-primary divisorial filtration, we have the Hilbert function f(n)=\lambda_R(R/I_n), where \lambda_R is the length of an R-module. We discuss how close or far this function is from being a polynomial, focusing on examples which are constructed and analyzed geometrically.
15:30 - 16:30 Wenbo Niu (University of Arkansas)
Title: Distinguished subvarieties of base locus of linear systems
Abstract: In this talk, we consider a finite rational map determined by a linear system with base locus. The degree of such map has been studied in many situation, for instance, the degree of Gauss map of theta divisors. In principal, this degree can be computed by Segre class of the base locus. In practice, one can use Vogel's cycle to give an estimation. Associated to the base locus, one can define distinguished subvarieties, which has been used to the study of geometric Nullstellensatz by Ein-Lazarsfeld. We discuss how distinguished subvarieties and their coefficients can be used to estimate the degree of finite rational map. This is a joint work with Yilong Zhang.
17:00 - 18:00 Hema Srinivasan (University of Missouri, Columbia)
Title: Kunz-Waldi semigroups and their generalizations
Abstract: Given two relatively prime positive integers, p < q, Kunz and Waldi defined a class of numerical semigroups which we denote by KW(p, q) consisting of semigroups of embedding dimension n and type n−1 and multiplicity p by filling in the gaps of the semigroup < a, b >. We study these semigroups, give a criterion for these in terms of principal matrices or their critical binomials and generalize the notion to KW(p, q, w) and prove some results and questions. We will discuss their resolutions and Betti Numbers. Most of this is a joint work with Srishti Singh.