Classifying algebraic varieties

Lines in the plane So, in most cases two lines intersect at a unique point... Lines in the plane So, in most cases two lines intersect at a unique point... Lines in the plane So, in most

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Higher dimensions Preview.

Classifying Algebraic Varieties I

Christopher Hacon

University of Utah

March, 2008

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2 equations in 2 variables

1 equation in 2 variables Irreducible subsets Curves Higher dimensions Preview.

Outline of the talk

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Outline of the talk

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Outline of the talk

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Outline of the talk

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Outline of the talk

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Outline of the talk

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Outline of the talk

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: : :

Pn(x1; : : : ; xm) = 0:

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1 equation in 1 variable

For example, 1 equation in 1 variable

P(x) = adxd+ ad 1xd 1+ : : : + a1x + a0 ad 6= 0

(a polynomial of degree d in x)

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P(x) = ad(x c1)(x c2) (x cd):

Therefore a polynomial of degree d always has exactly d

solutions or roots (when counted with multiplicity)

If one is interested in solutions that belong to R (or Q, or Zetc.), then the problem is much more complicated, but atleast we know that there are at most d solutions

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P(x) = adxd+ ad 1xd 1+ : : : + a1x + a0 ad 6= 0(a polynomial of degree d in x)

P(x) = ad(x c1)(x c2) (x cd):

Therefore a polynomial of degree d always has exactly dsolutions or roots (when counted with multiplicity)

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Complex solutions

Throughout this talk I will always look for complex solutions(z1; : : : ; zm) 2 Cm to polynomial equations

P1(x1; : : : ; xm) = 0 : : : Pn(x1; : : : ; xm) = 0where Pi 2 C[x1; : : : ; xm]

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Outline of the talk

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Lines in the plane

Consider now 2 equations in 2 variables

For example 2 lines in the plane i.e

There are 3 cases

1 If the lines coincide (i.e if P1(x; y) = cP2(x; y) for some

0 6= c 2 C), then there are in nitely many solutions

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Lines in the plane

There are 3 cases

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Lines in the plane

There are 3 cases

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Lines in the plane

There are 3 cases

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Lines in the plane

There are 3 cases

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Lines in the plane

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Lines in the plane

So, in most cases two lines intersect at a unique point

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Lines in the plane

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Lines in the plane

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Lines in the plane

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Lines in the plane

We may thus think of two distinct parallel lines as meeting in

exactly one point at in nity and we have

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Lines in the plane

We may thus think of two distinct parallel lines as meeting inexactly one point at in nity and we have

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Lines in the projective plane

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The projective plane

To make things precise, one de nes

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To make things precise, one de nes

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To make things precise, one de

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