The configuration of bitangents of the Klein curve, R.H. The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson, William Thurston The Eightfold Way: The Beauty of Klein's Quartic Curve, Silvio Levy, ed. Bezout's theorem and its applications, Geunho Gim Euclid meets Bezout: Intersecting algebraic plane curves with the Euclidean algorithm, Jan Hilmar and Chris Smyth Solving the cubic and quartic, Aaron Landesman Foundations of Algebraic Geometry, Ravi Vakil Some naive enumerative geometry, James McKernan Hilbert's Nullstullensatz, Daniel Allcock Undergraduate Algebraic Geometry, Miles Reid Algebraic Geometry I, Karen Smith (notes by David Bruce) An Invitation to Algebraic Geometry, Karen Smith et al. Introduction to Projective Varieties, Enrique Arrondo Undergraduate Algebraic Geometry, Joe Harris (notes by Aaron Landesman) Resources - Algebraic Geometry, Andreas Gathmann Describe an order-preserving bijection between the primes of S -1R and the primes of R that don't meet S. Let R be a ring and S a multiplicative subset.Using primality, show that one of the linear factors of h, say x-a, is in p. Using the Euclidean algorithm in the Euclidean domain C(x), show that you can find a nonzero h in (f,g) < p. Show you can find f,g in p with no common factor. Show that the only non-principal prime ideals in C are of the form (x-a,y-b).(c) For every complex number a, the ideal (x-a,y-a) contains the ideal (x-y). (b) For two fixed complex numbers a and b, show that (x-a,y-b) is a maximal ideal in R and hence is also a point in Spec R. (a) Show that (x-y) is a prime ideal in R, and hence is a point in Spec R. Show that a morphism of affine algebraic varieties induces a linear map on each tangent space.Show that a point of V is singular if and only if it is a singular point of Z(f), Z(g), or their intersection. Assume that f and g are coprime elements of k and let V=Z(fg).Find conditions on a and b that determine whether or not the given curve is smooth. Consider the curve in A 2 given by y 2=x 3+ax+b, where a and b lie in k.Prove the classification of conics in P 2.Verify the defining equations for the images of the Veronese maps.Describe the images of the lines in P 2 in the Veronese surface. Its image is called the Veronese surface. Consider the Veronese map ν 2 : P 2 → P 5.Show that the induced topology is not the product topology, assuming that neither of the varieties in the product is a finite collection of points. Define the topology on a product of varieties by declaring the Segre map to be an isomorphism of P n x P n onto V m,n.Assuming that the Segre map is an isomorphism of P n x P n onto V m,n, show that the diagonal in P n x P n is a closed set. Find bihomogeneous polynomials describing the twisted cubic as a subset of P 1 x P 1. Show that the twisted cubic lies in the Segre variety V 1,1. Regard the twisted cubic as the image of the map P 1 → P 3 given by →.Show that the product of two affine varieties is an affine variety. Define the product of two affine varieties to be the image under the appropriate Segre map.(a) Show that to linear subspaces of P (m+1)(n+1). For each f in k, let U f be the set of points p in X so that f(p) is nonzero. Let X be an affine algebraic variety in A n and let k be its coordinate ring, where k is algebraically closed.Prove the uniqueness statement: If f : X → Y and g : X → Y are maps with f * = g * then f=g.Show that f * : k → k is surjective if and only f f defines an isomorphism between X and a subvariety of Y.Show that f * : k → k is injective if and only if f is dominant, that is, theimage f(X) is dense in Y.
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