Workshop on Algebraic Geometry

19-21 November 2019              KIAS 1503

Title&Abstract Home > Title&Abstract

 

Marian Aprodu (Bucharest)

- Title: Green's conjecture and vanishing of Koszul modules

- Abstract: I report on a joint with Gavril Farkas, Stefan Papadima, Claudiu Raicu and Jerzy Weyman. Koszul modules are multi-linear algebra objects associated to an arbitrary subspace in a second exterior power. They are naturally presented as graded pieces of some Tor spaces over the dual exterior algebra. Koszul modules appear naturally in Geometric Group Theory, in relation with Alexander invariants of groups. We prove an optimal vanishing result for the Koszul modules, and we describe explicitly the locus corresponding to Koszul modules that are not of finite length. We use representation theory to connect the syzygies of rational cuspidal curves to some particular Koszul modules and we prove that our vanishing result is equivalent to the generic Green conjecture.

 

Thomas Dedieu (Toulouse)

- Title: Extensions of canonical curves and K3 surfaces

- Abstract: Given a variety X in a projective space P^n, one asks whether it has extensions, i.e., varieties Y in P^{n+k}, not cones, such that X is a linear section of X.
When X is a genus g curve C canonically embedded in P^{g-1}, the possible smooth extensions of C in P^g and P^{g+1} respectively are K3 surfaces and Fano manifolds. A recent result of Arbarello--Bruno--Sernesi says that essentially such a curve C is extendable to a surface if and only if a certain cohomological map (the so-called Gaussian map) is non-surjective.
During the talk I will report about joint work with C. Ciliberto and E. Sernesi. I shall explain how to generalize the former result to the extendability of C to higher dimensional varieties, then give various applications, in particular a necessary and sufficient condition for the extendability of K3 surfaces.

 

Baohua Fu (Beijing)

- Title: Rigidity of wonderful compactifications under Fano deformations

- Abstract: For an adjoint simple Lie group $G$, there exists a unique $G times G$-equivariant compactification $X$ of $G$, which is Fano of Picard number equal to the rank of $G$ and it enjoys many interesting properties. In a joint work with Qifeng Li, we will show that such wonderful compactifications are rigid under Fano deformations except possibly for $G$ of type $B_3, G_2, F_4$ and  $E_8$.

 

Yong Hu (KIAS)

- Title: On Donagi-Markman cubic for Lagrangian fibration induced by VMRT of cubic 4-folds.

- Abstract: Let $X$ be a smooth cubic $4$-fold. It is well known that $X$ is covered by lines. Let $mathcal{K}$ be the family of lines on $X$.   Denote by $xin X$ a general point and by $mathcal{K}_x$ the lines passing through $x$. The subvariety $mathcal{C}_xsubsetmathbb{P}(T_xX)$, defined as the image of the tangent map $tau_x$ which sends a member of $mathcal{K}_x$ to its tangent direction at $x$, is called the variety of minimal rational tangents (VMRT) at $x$. On a Zariski open subset $U$ of $X$, for every $xin U$, the VMRT $mathcal{C}_x$  is a smooth projective curve of genus $4$. Let $mathcal{C} to U$ be the smooth fibration induced by VMRT of $X$ . I will present a formula for the Donagi-Markman cubic for the Jacobian fibration induced by $mathcal{C} to U$. This is a work in progress.

 

Zhi Jiang (Shanghai)

- Title: Decomposition theorems for the pushforwards of pluricanonical bundles to abelian varieties

- Abstract: We will discuss M-regularity decomposition theorems of pushforwards of pluricanonical bundles or adjoint bundles to abelian varieties and explain some geometric applications.

 

Jie Liu (Beijing)

- Title: Fano manifolds containing locally rigid Fano divisors

- Abstract: It is a very classical problem in adjunction theoty to ask which ambient manifolds can be determined by the existence of some special divisors. In this talk, I will consider the case where X is a Fano manifold containing a locally rigid Fano divisor A with Picard number one. In particular, I will explain how to use the recent progress of VMRT theory to translate the problem into projective geometry. Moreover, if the time permits, I will present an application to the classification of Fano manifolds X containing a rational homogeneous space as exceptional divisor.

 

Shouhei Ma (Tokyo)

- Title: Mukai models and Borcherds products

- Abstract: We give a correspondence between pluricanonical forms on the moduli space F_{g,n} of n-pointed K3 surfaces of genus g with at worst rational double points and orthogonal modular forms of weight divisible by 19+n, twisted by the determinant character and with vanishing condition at the (-2)-Heegner divisor. Then we give application to the Kodaira dimension of F_{g,n}. We use Borcherds products to find a lower bound of n where F_{g,n} has nonnegative Kodaira dimension, and compare this with an upper bound where F_{g,n} is unirational or uniruled using classical and Mukai models in g<21.

 

Eyal Markman (Amherst)

- Title: The Hodge conjecture for the generic abelian fourfold of Weil type with discriminant 1.

- Abstract: A generalized Kummer variety of dimension 2n is the fiber of the Albanese map from the Hilbert scheme of n+1 points on an abelian surface to the surface. We compute the monodromy group of a generalized Kummer variety via equivalences of derived categories of abelian surfaces.
As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but with trivial discriminant invariant. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type.
If time permits we will discuss a second application, the proof of the Generalized Hodge Conjecture for complex codimension two cycles on every projective generalized Kummer variety (i.e., surjectivity of the Abel-Jacobi map onto its third intermediate Jacobian).

 

Shigeru Mukai (Kyoto/KIAS)

- Title: Automorphism groups of maximal supersingular K3 surfaces over the prime fields

- Abstract: The surface in the projective 5-space defined by three symmetric polynomials of degree 1, 2, 3 is a K3 surface of degree 6 over the ring of integers. The mod p reduction is supersingular if and only if p divides the discriminant (= -120) or the discriminant is non-residue. A system of generators of the group of automorphisms over the prime field F_p is constructed for the supersingular reduction of the K3 surface in the former case p = 2, 3 and 5, where the number of F_p-rational points attains the maximum (= p^2+20p+1).

 

Jinhyung Park (Sogang Univ.)

- Title: Singularities and syzygies of secant varieties of a nonsingular projective curve

- Abstract: It is a classical result of Castelnuovo, Mumford, and many others that a nonsingular projective curve embedded by a very ample line bundle with sufficiently large degree is projectively normal and the defining ideal is generated by quadrics. Green realized that the classical questions on defining equations should be generalized to higher syzygies, and proved his famous (2g+1+p)-theorem. In this talk, we show that the k-th secant variety of the curve are arithmetically Cohen-Macaulay and satisfies the property N_{k+2,p}. This result was conjectured by Sidman-Vermeire, and it may be regarded as a generalization of Green's theorem. We also prove that the higher secant varieties have normal Du Bois singularities. This result was conjectured by Ullery. This is joint work with Lawrence Ein and Wenbo Niu.

 

Gregory Sankaran (Bath)

- Title: Families of generalised Kummer manifolds

- Abstract: There are several geometric constructions that yield irreducible holomorphic symplectic manifolds, but in almost all cases they are of the deformation type of a Hilbert scheme of a K3 surface. I will describe a possible approach (work in progress) to constructing a family of manifolds of generalised Kummer type.

 

Justin Sawon (Chapel Hill)

- Title: Lagrangian fibrations by Prym varieties

- Abstract: Lagrangian fibrations are fibrations on holomorphic symplectic manifolds/orbifolds whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic structure. Beauville and Mukai considered examples coming from relative Jacobians of families of curves on K3 surfaces. We discuss the construction of examples whose fibres are Prym varieties, coming from K3 surfaces covering del Pezzo surfaces (following Markushevich-Tikhomirov, Menet, Matteini). Some interesting duality relations arise in these examples.

 

Gerard van der Geer (Amsterdam)

- Title: Modular forms and invariant theory

- Abstract: Siegel and Teichmueller modular forms of genus g are generalizations of the usual elliptic modular forms, the case g=1, but live on the moduli space of abelian varieties and curves. These forms are just as intriguing, but more difficult to construct. We intend to show how one can use invariant theory to describe such modular forms for degree 2 and 3 explicitly. This is joint work with Fabien Clery and Carel Faber.