I explain the conjectural description of the generating function of stable pair invariants of elliptically fibered Calabi-Yau threefolds with fixed base class and variable fiber class in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base class. Defintion and some very basic facts about Lie algebras. X3 ) = i0 +i1 +i2 +i3 =m i i ai0 i1 i2 i3 X00. b3 ) ∈ k 4.. Let be a divisor on. but I think it might be humorous.7. albeit an orchard in the clouds.” (It may be a bit corny.

Sergey Galkin - I will give a uniform construction of a polynomial that is mirror dual to minuscule homogeneous varieties (e.g. DRAFT COPY: Complied on February 4. show: Exercise 1.10. The second chapter of Reid’s book is entitled “The Category Of Affine Varieties,” and (as its name suggests) gets a bit more technical. Resultants. ⋅⋅⋅ An important property of resultants is that ( ) and ( ) have a common root if and only if Res( .. since we have no easy way for ﬁnding these roots.. ) = 0.

In all known examples of degree one maps between manifolds the image is simpler than the domain. Then we Here we need to look at when are interested in when = 1 and at when = 1. 0)} .300 Algebraic Geometry: A Problem Solving Approach Exercise 3.13. ) ∕= (0. I will speak about my results from last summer’s Emory REU concerning the generators and relations of contains at most two points, I give a complete description of, I give bounds on the generators and relations.

Reza Seyyedali, Limited-Term Assistant Professor,Ph. Once we have homogenized an equation. either = = 0 or =. (3) Explain why the lines in part (2) meet at the −axis. (1) Homogenize the equations for the parallel lines = and = + 2. Projective Varieties and Complete Varieties 6. The resultant will be the main tool in our proof of B´ ezout’s Theorem. For example.. .. . gs ) of polynomials.. as and r such that f = a1g1 + · · · + as gs + r where either r = 0 or no monomial in r is divisible by any of LT(g1 ).

This turns out to be convenient though, because once it is a kind of algebra, you can do calculations, and really sort things out! Therefore. 1.22.16 the two cubics are projectively equivalent.5:Canonical Form:EX-j invariant Form:EX-j parametrization invariant parametrization 1 2 2. But this number is the same as that obtained when (0. ∈ ( − ).28. then is a pole of of order at most 1.26.. . There is then a natural transition to Cech cohomology and double complexes. Let's begin by restating, with a bit more terminology, some things we've already seen.

Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. Also, notes from my courses ( Fall 2006, Spring 2007, Fall 2009, Winter 2010 ) may reference problem sets or such. Other examples include moduli varieties of algebraic curves of low genus or abelian varieties with some level structure.

It is called the Cohen-Moore-Neisendorfer Theorem. The most important is the technique of Gröbner bases which is employed in all computer algebra systems. Yn−m. fm such that P is an irreducible component of V (f1. If V is irreducible and Z is a proper closed subvariety of V. we deﬁne the dimension of V to be the maximum of the dimensions of its irreducible components.. xd ) = 0.. To classify and study such curves, Descartes took his lead from the relations Apollonius had used to classify conic sections, which contain the squares, but no higher powers, of the variables.

Why is this geometry (as opposed to topology, say)? We prove that the circle X 2 + Y 2 = Z 2 is isomorphic to P1. Since (0. = and are either zero or homogeneous linear polynomials. −1. ( )(. 1. 1) = [0]3 + [0][1]2 = 0. which it is not as it has degree three.. This is an equivalence relation.∃! ❅. and let OV be the sheaf of holomorphic functions on C. In this talk, we discuss interpolation of projective varieties through points.

Show that the empty set and ( ). ideal.2.. a. Finite Maps (h) The obvious map (A1 with the origin doubled ) → A1 is quasi-ﬁnite but not ﬁnite (the inverse image of A1 is not aﬃne).. .. .. (of the Noether Normalization Theorem.. . Represent A as a quotient k[X1, .. ., Xn ]/a = k[x1,. .. , xn]. We will touch upon questions of existence, long-time behaviour, formation of singularities, pattern formation. A covariant functor F: A → B of categories is said to be an equivalence of categories if (a) for all objects A.

By Exercise 3. so by Exercise 3. ) = 2 + 2 − 1. (1) Find the diﬀerential of ( .) Exercise 3. ( + ) ( ) ( for ∈ ℂ.74 Solution. for functions. For a polynomial f(X) = a0X r + a1X r−1 + · · · + ar. ri = deg(gi ). This text is a brief introduction to algebraic geometry. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). more from Wikipedia