Email Address: mim2 (at) illinois (dot) edu [OR]
mim2 (at) math (dot) uiuc (dot) edu
Mailing Address: 250 Altgeld Hall, 1409 West Green Street, Urbana, IL 61801
Office Address: 223 Illini Hall, 725 South Wright Street, Champaign, IL 61820 Location
Office Phone Number: 001-217-333-1898
Office Hours: Mondays 1-2PM in 223 Illini Hall during Spring 2010
Enterprise: To sign up for classes


I ran the Graduate Student Algebraic Geometry Working Seminar during Spring of 2010. I will not be running this seminar in Fall of 2010 but I may continue this seminar in the Spring of 2011.


Go here for a list to all the math seminars at U of I and go here to go to the U of I Mathematics calendar.




Algebraic Geometry Graduate Student Working Seminar for Spring 2010
We meet on Tuesdays and/or Thursdays in room 347 in Altgeld Hall.

• Understanding First Order Deformation (Part 1) January 26, 2010
  • Speaker: Steve Maguire
  • Time: 1PM-2PM
  • Abstract: We will concentrate on deformation theory and its applications to moduli theory, Mori Program, and birational geometry this semester. We are using Robin Hartshorne's book titled Deformation Theory. I will give the definition of deformation, talk about why we should care about such a thing, and give a proof that H⁰(Y, N_Y/X) parametrizes the deformation of a closed subscheme Y in X.
    
    
• Understanding First Order Deformation (Part 2) February 2, 2010
  • Speaker: Mee Seong Im
  • Time: 1PM-2PM
  • Abstract: This is the second of a series of lectures from Robin Hartshorne's Deformation Theory. I will discuss structures such as a (sub)scheme flat over the dual numbers (this is thought of as an infinitesimal thickening a scheme evenly spread out over the base), some applications of the Tⁱ functors (e.g., given a k-algebra B where k is a field, deformations of B over the dual numbers are classified by T¹(B/k,B) and its obstructions lie in T²(B/k,B)), the infinitesimal lifting property, and deformation of rings (this has to do with finding various extensions of a ring B by a B-module). I will also talk about when trivial deformations and trivial extensions are the only possibility for a scheme. Since this is a seminar for graduate students, I will attempt to provide as many examples as possible during the lecture.
    
    
• Higher Order Deformation (Part 3) February 9, 2010
  • Speaker: Steve Maguire
  • Time: 1PM-3PM
  • Abstract: I will talk about higher order deformations and obstruction theory applied to subschemes, invertible sheaves, vector bundles and coherent sheaves. Jimmy Shan and I will prove when a locally free sheaf over a scheme X can be deformed when X is deformed, the necessity and the sufficiency of the vanishing of its obstruction for the existence of global deformation, and which group the automorphism of the deformed sheaf is isomorphic to. The first hour will be devoted to lecture while the second hour (2-3PM) will be devoted to working on problems from Hartshorne's book.
    
    
• Higher Order Deformation (Part 4) February 16, 2010
  • Speaker: Mee Seong Im
  • Time: 1PM-3PM
  • Abstract: We will understand how deformations of a Cohen-Macaulay subscheme Y of codimension 2 in X relate to a matrix whose entries are from the coordinate ring of the ambient space X. We will analyze this from a regular local ring perspective, by taking X to be affine (we can reduce this case to the local case by intersecting our CM subscheme of codimension 2 with an affine open set in X), and then by taking X to be a projective scheme. When Y is closed and projective, we need to assume that Y is arithmetically Cohen-Macaulay in codimension 2. Then similar deformation results arise in the projective case whenever the dimension of Y is strictly greater than 0. We briefly analyze deformation for a complete intersection and for Gorenstein rings in codimension 3.
    
    
• Higher Order Deformation (Part 5) February 23, 2010
  • Speaker: Mee Seong Im
  • Time: 1PM-3PM
  • Abstract: We will begin by Jimmy Shan presenting the following paper: Linearizing Flows and a Cohomological Interpretation of Lax Equations by P.A. Griffiths. He will cover Hamiltonian System/Lax Equations, Spectral Curves, Eigenvector Mapping, and Griffiths' Linearizing Theorem. We will then continue with our study of deformation theory from Hartshorne's book. We will study obstructions to deformations of schemes, dimensions of families of spatial curves, and analyze a nonreduced component of the Hilbert scheme. This concludes Chapter 2 of Hartshorne. We will spend the next few weeks carefully studying Compact Complex Surfaces by Wolf Barth and Higher-Dimensional Algebraic Geometry by Olivier Debarre. If we have time, we will return to moduli theory and global deformation theory in Hartshorne's book.
    
    
• Please note that our study of Compact Complex Surfaces by Wolf Barth has been delayed for a few weeks.
  
  
• Higher-Dimensional Algebraic Geometry (Part 1) March 2, 2010
  • Speaker: Steve Maguire
  • Time: 1PM-3PM
  • Abstract: I will give a lecture on Chapter 2 from Olivier Debarre's book titled Higher-Dimensional Algebraic Geometry. Chapter 2 covers parametrizing morphisms. We will first learn to parametrize all rational curves from P¹ to P^N. These morphisms of degree d form a quasi-projective variety Mor_d(P¹, P^N). We learn that these morphisms fit together to give us a universal morphism. We then obtain Mor(P¹, P^N), which is a locally Noetherian disjoint union of Mor_d(P¹, P^N) for d ≥ 0. We then generalize to the space Mor(Y,X) when Y is a projective variety and X is quasi-projective. We state and prove that the tangent space to Mor(Y,X) at [f] is isomorphic to H⁰(Y,Hom(f^*Ω_X, O_Y)). We then look at the local structure of Mor(Y,X) where X and Y are both projective. We begin the proof that Mor(Y,X) at [f] is locally defined by h¹(Y,f^*T_X) equations in a nonsingular variety of dimension h⁰(Y,f^*T_X). I am left to prove that the dimension of any irreducible component of Mor(Y,X) through the point [f] is at least h⁰(Y,f^*T_X) - h¹(Y,f^*T_X). We will continue with this section next week. All graduate students are welcome to attend as this is a friendly working environment.
    
    
• Higher-Dimensional Algebraic Geometry (Part 2) March 11, 2010 in 1 Illini Hall
  • Speaker: Steve Maguire
  • Time: 1PM-3PM
  • Abstract: This seminar HAS BEEN moved to Thursday. We will finish Chapter 2 from Debarre's book. We will finish our study of the local structure of Mor(Y,X). We will then go into parametrizing morphisms with fixed points, of flat families, from a curve, and from a curve over a base. We will then study lines on a subvariety of a projective space over an algebraically closed field.
    
    
• Higher-Dimensional Algebraic Geometry (Part 3) March 18, 2010 in 1 Illini Hall
  • Speaker: Mee Seong Im
  • Time: 1PM-3PM
  • Abstract: Chapter 1 in Debarre's book consists of divisors and 1-cycles, intersection numbers, cone of curves, Nakai-Moishezon Ampleness Criterion, nef divisors, an asymptotic form of Riemann-Roch, and rational curves on exceptional loci of a morphism. A Cartier divisor is a global section of the sheaf K^*/O^* and a Weil divisor is a finite linear integral combination of integral hypersurfaces in X. Given a subscheme X of dimension n in the projective N-space with X being of finite type over k, each m → χ(X,O_X(m)) ∈ Z. That is, χ(X,O_X(m)) is the Hilbert polynomial of X of degree n with rational coefficients with deg(X)=n!(coefficient of mⁿ). We then generalized this to r Cartier divisors on a proper scheme X: let D₁,..., D_r be Cartier on a proper scheme X. Let F be a coherent sheaf on X. Then (m₁,...,m_r) → χ(X,F(m₁D₁+...+m_rD_r)) takes values on Z^r as a polynomial with rational coefficient of degree at most the dimension of supp(F). Note that the leading term of the Hilbert polynomial is 0 when r > dim X. We then stated the projection formula and then generalized this for r Cartier divisors: given a surjective morphism π: Y → X between two proper varieties and Cartier divisors D₁,..., D_r on X with r > dim Y, we have π^*D₁...π^*D_r = deg(π)(D₁...D_r). We came up with a few examples for the above and worked on an exercise from Debarre. We concluded our seminar by studying the cone of curves. For X a proper scheme, we say two Cartier divisors D and D' on X are numerically equivalent if they have the same degree on each curve. We write D ~ D'. We denote the group of Cartier divisors mod the numerically equivalent classes as N¹(X)_Z. We can also work with N¹(X)_Q and N¹(X)_R by tensoring N¹(X)_Z with Q or R, respectively. All three are finite dimensional vector spaces and we call their dimension the Picard number of X. Two 1-cycles (curves) C and C' are numerically equivalent if they have the same intersection number with every Cartier divisor. We write C ~ C'. We write this quotient group as N₁(X)_Z. The intersection pairing between N¹(X)_Z and N₁(X)_Z gives us a map into Z as a nondegenerate pairing. N₁(X)_R contains convex cone of curves NE(X), which is the set of classes of effective 1-cycles. NE(X) also contains NE(π) where π: X → Y is a map of projective varieties. NE(π) is generated by the classes of curves contracted by π. We conclude that NE(X) is extremal if a, b ∈ NE(X) and a+b ∈ NE(π), then a, b ∈ NE(π). We will continue with Chapter 1 after the break.
    
    
• Spring Break: March 22 to March 26, 2010
  
  
• Higher-Dimensional Algebraic Geometry (Part 4) April 1, 2010 in 1 Illini Hall
  • Speaker: Mee Seong Im
  • Time: 1PM-3PM
  • Abstract: We reviewed the general Hilbert polynomial when given r Cartier divisors on a proper scheme X, the projection formula for r Cartier divisors on X, numerical equivalence for Cartier divisors and for 1-cycles, N¹(X) = the set of Cartier divisors on X/~, N₁(X) = the set of 1-cycles on X/~, convex cone NE(X) of curves consisting of classes of effective 1-cycles sitting inside N₁(X)_R, and relative cone NE(π) of a morphism π: X → Y between two projective schemes X and Y which is generated by classes of curves contracted by π. NE(π) is closed and convex and because of Stein factorization, we have NE(π) = NE(π') where π':X → Y' has connected fibers and Y' → Y is finite. Being contracted is a numerical property and we see that NE(π) has the property of being extremal. This raises the important question of characterizing extremal subcones of NE(X) that correspond to morphisms. We concluded by discussing Nakai-Moishezon and Kleiman's criterion and nef and big Cartier divisors.
    
    
• Higher-Dimensional Algebraic Geometry (Part 5) April 8, 2010 in 1 Illini Hall
  • Speaker: Steve Maguire
  • Time: 1PM-3PM
  • Abstract: I will start Chapter 3 in Debarre's book titled "Bend-and-Break" lemmas. Given f: C → X a morphism from a smooth curve to a projective variety, we will learn to produce rational curves on X through f(c) when dim_[f] Mor(C,X; f|_{c}) ≥ 1. After proving this proposition, we will discuss rational curves on Fano varieties, stronger and relative version of bend-and-break lemma, and rational curves on varieties whose canonical divisor is not nef. We will continue our study of Chapter 3 next week.
    
    
• Higher-Dimensional Algebraic Geometry (Part 6) April 9, 2010, meet in the common room and go to an empty classroom
  • Speaker: Mee Seong Im
  • Time: 12:30PM-2PM
  • Abstract: We will finish Chapter 1 of Debarre. We will restate Nakai-Moishezon and Kleinman's criterion and prove them, prove that the sum of a nef and ample divisor on a projective scheme is ample while the sum of two nef divisors is nef, and conclude that ample classes in N¹(X)_Q form an open cone while the closure of this cone gives us a nef cone. We define when a nef divisor is big and also give the definition of when a Cartier divisor is big without requiring it to be nef, analyze an asymptotic form of Riemann-Roch for any Cartier divisor, discuss exceptional locus of a morphism and rational curves on exceptional loci. We will give examples of cone of curves that are 1-dimensional, on the product of projective spaces, on ruled and cubic surfaces, and cone of curves that is not finitely generated. We conclude by understanding the problem of classification of algebraic varieties and how rational curves play a crucial role.
    
    
• Higher-Dimensional Algebraic Geometry (Part 7) April 13, 2010, 347 Altgeld Hall (meet in the common room if we're not there)
  • Speaker: Mee Seong Im
  • Time: 12PM-2PM
  • Abstract: Chapter 3 of Debarre: continue with Bend-and-Break lemmas with concrete examples. We reproved one of the first Bend-and-Break lemmas: assuming X to be a projective variety and letting f to be the morphism from a curve C to X with c ∈ C, if dim_[f]Mor(C,X; f|_c) ≥ 1, then there exists a rational curve on X through f(c). We then quickly reviewed other versions of Bend-and-Break lemmas.
    
    
• Higher-Dimensional Algebraic Geometry (Part 8) April 15, 2010, 347 Altgeld Hall (the first hour), 341 Altgeld Hall (the second hour)
  • Speaker: Mee Seong Im, Steve Maguire
  • Time: 12PM-2PM
  • Abstract: The first hour will be devoted to Bend-and-Break lemmas. If X is not covered by rational curves, we saw that the minimal model X₀ for X is unique and the canonical sheaf associated to X₀ is nef. If X is covered by rational curves, we saw that the minimal model for X is not unique and K_X0 is not nef. Examples of a variety that is covered by rational curves are Fano varieties. We saw X is Fano if X is smooth, projective, and its anticanonical sheaf is ample. They include P¹ x P¹, P², Pⁿ, and the product of Fano varieties. We studied the exceptional locus of a morphism and that a contractible rational curve passes through the general point of each component of the exceptional locus. We also studied relative Bend-and-Break lemma. It says that given X and a morphism π: X → Y deforming a curve C in X while fixing a point on C can have one of two possibilities-- 1.) the images of C in Y vary or 2.) the images in Y do not vary. For case 1, we produce rational curves in X not contracted by π and for case 2, we produce rational curves in X contracted by π. We conclude that for Fano varieties, there are rational curves transverse to any nonconstant X → Y. Thus any two points of a Fano can be joined by a chain of rational curves. The second hour is devoted to working on higher-dimensional algebraic geometry and deformation theory problems led by Steve Maguire. We worked on a problem about linkage from Hartshorne's Deformation Theory text. Prepare to apply what we have been discussing this semester.
    
    
• Higher-Dimensional Algebraic Geometry (Part 9) April 20, 2010, 347 Altgeld Hall (meet in the common room if we're not there)
  • Speaker: Mee Seong Im
  • Time: 12PM-2PM
  • Abstract: Review and summarize chapter 3 of Debarre. Reprove the relative bend-and-break lemma. We will then study uniruled and rationally connected varieties from chapter 4 of Debarre. We saw from chapter 3 that Fano varieties are covered by rational curves which implies that they do not have unique minimal model and the canonical sheaf associated to the minimal model is not nef. For example, P¹ x P¹ and P² are birational but not isomorphic. We can get P¹ x P¹ from P² by blowing up two points on P² (call this variety X) and then blow down the line connecting the two exceptional divisors to obtain P¹ x P¹. P² and P¹ x P¹ are both nonsingular and smooth and since they do not have any curves with self-intersection number -1, we have two nonisomorphic minimal surfaces for X. A surface covered by rational curves is called unirational. We finish by understanding the notion of uniruled, free, rationally (chain-)connectedness, and very free and then come up with examples for each.
    
    
• Higher-Dimensional Algebraic Geometry (Part 10) April 22, 2010, 347 Altgeld Hall (the first hour), 341 Altgeld Hall (the second hour)
  • Speaker: Steve Maguire
  • Time: 12PM-2PM
  • Abstract: The first 30 minutes to an hour will be devoted to lecture from Debarre. The rest of the seminar will be devoted to higher-dimensional algebraic geometry and deformation theory problems. Prepare to apply what we have been discussing this semester. We may spend up to an hour this semester watching one of the MSRI lecture videos from Spring 2009 that is related to what we have been studying.
    
    
• Higher-Dimensional Algebraic Geometry (Part 11) April 27, 2010, 347 Altgeld Hall
  • Speaker: Mee Seong Im
  • Time: 12PM-2PM
  • Abstract: Chapter 4.
    
    
• Higher-Dimensional Algebraic Geometry (Part 12) April 29, 2010, 347 Altgeld Hall
  • Speaker: Mee Seong Im or Steve Maguire
  • Time: 12PM-1PM
  • Abstract: Flips and flops.
    
    
• Higher-Dimensional Algebraic Geometry (Part 13) May 4, 2010, 243 Altgeld Hall
  • Speaker: Mee Seong Im
  • Time: 12PM-2PM
  • Abstract: To be announced. Last meeting for the semester. We may continue sporadically over the summer. This meeting was canceled due to everyone's busy schedule. Have a great summer!
    
    

U of I Math Tutoring Services Educational Services-Counseling Center