Algebraic Geometry: Curves, Varieties, and Riemann Surfaces

The projective space $\mathbb{P}^n$ is the set of all lines through the origin in $\mathbb{R}^{n+1}$. It has dimension $n-1$.

The degree of a polynomial is the highest exponent of the variable that appears in the polynomial. Since the zero polynomial has no variable, its degree is undefined.

The equation of the tangent line to a curve at a point is given by the formula $y - y_1 = m(x - x_1)$, where $m$ is the slope of the tangent line and $(x_1, y_1)$ is the point of tangency. In this case, the slope of the tangent line is $m = f'(1) = 3(1)^2 - 2(1) + 1 = 2$. So the equation of the tangent line is $y - (-1) = 2(x - 1)$, which simplifies to $y = 3x - 4$.

The genus of a Riemann surface is a topological invariant that is related to the number of holes in the surface. In this case, the Riemann surface defined by the equation $y^2 = x^3 + x + 1$ has one hole, so its genus is 1.

The Hilbert polynomial of a ring is a polynomial that gives the number of generators of the ring in each degree. In this case, the ring $\mathbb{C}[x, y]/(x^2 + y^2)$ is generated by the elements $x$ and $y$, so its Hilbert polynomial is $t^2$.

The dimension of a variety is the number of independent variables that are needed to parameterize it. In this case, the variety defined by the equation $x^2 + y^2 + z^2 = 1$ is a sphere, which is a two-dimensional surface. So its dimension is 2.

The degree of a divisor on a Riemann surface is the sum of the orders of its zeros minus the sum of the orders of its poles. In this case, the divisor $(x-1)(x-2)$ has two zeros at $x=1$ and $x=2$, and no poles. So its degree is $2$.

The Picard group of an elliptic curve is a group that is related to the number of linearly independent holomorphic differentials on the curve. In this case, the elliptic curve $y^2 = x^3 + x + 1$ has one linearly independent holomorphic differential, so its Picard group is $\mathbb{Z}$.

The genus of a curve is a topological invariant that is related to the number of holes in the curve. In this case, the curve defined by the equation $y^2 = x^5 + x^3 + x$ has one hole, so its genus is 1.

The degree of a divisor on a Riemann surface is the sum of the orders of its zeros minus the sum of the orders of its poles. In this case, the divisor $(x-1)^2$ has two zeros at $x=1$, and no poles. So its degree is 2.

The Picard group of an elliptic curve is a group that is related to the number of linearly independent holomorphic differentials on the curve. In this case, the elliptic curve $y^2 = x^3 + 2x + 1$ has two linearly independent holomorphic differentials, so its Picard group is $\mathbb{Z}/2\mathbb{Z}$.

The genus of a curve is a topological invariant that is related to the number of holes in the curve. In this case, the curve defined by the equation $y^2 = x^6 + x^4 + x^2$ has two holes, so its genus is 2.

The degree of a divisor on a Riemann surface is the sum of the orders of its zeros minus the sum of the orders of its poles. In this case, the divisor $(x-1)^3$ has three zeros at $x=1$, and no poles. So its degree is 3.

The Picard group of an elliptic curve is a group that is related to the number of linearly independent holomorphic differentials on the curve. In this case, the elliptic curve $y^2 = x^3 + 3x + 2$ has three linearly independent holomorphic differentials, so its Picard group is $\mathbb{Z}/3\mathbb{Z}$.