Return to Answer

First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either

a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomologically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.

Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.

Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.

Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.

Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.
In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.

Edit (added by Charles with Georges's permission): Fulton's axiomatic characterization may be described as follows: There is a unique assignment of a number $\mathcal{A}(X)$ to every [smooth irreducible projective variety over a fixed algebraically closed field] (hereafter simply "variety"), such that the following three axioms are satisfied:

1. $\mathcal{A}$ respects isomorphism classes.

2. If $X$ is a point, then $\mathcal{A}(X) = 1$.

3. Let $X$, $Y$, and $Z$ be (smooth) varieties of the same dimension. Suppose that $X$, $Y$, and $Z$ can be embedded as codimension-one subvarieties of a common (smooth) variety $W$, such that

   • $X$ and $Y+Z$ are linearly equivalent as divisors in $W$, and
   • $Y$ and $Z$ intersect transversely in a disjoint union of (smooth) varieties $V_1, \dotsc, V_{\ell}$.

   Then $$\mathcal{A}(X) = \mathcal{A}(Y) + \mathcal{A}(Z) - \sum_i \mathcal{A}(V_i).$$

This assignment takes $X$ to its "Hirzebruch arithmetic number" $\mathcal{A}(X) = \chi(X, \mathcal{O}_X)$.

First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either

a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomologically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.

Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.

Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.

Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
[This formula (and others) can be found in Hartshorne, Chapter I, Exercise 7.2, page 54]
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.

Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.
In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.

Edit (added by Charles with Georges's permission): Fulton's axiomatic characterization may be described as follows: There is a unique assignment of a number $\mathcal{A}(X)$ to every [smooth irreducible projective variety over a fixed algebraically closed field] (hereafter simply "variety"), such that the following three axioms are satisfied:

1. $\mathcal{A}$ respects isomorphism classes.

2. If $X$ is a point, then $\mathcal{A}(X) = 1$.

3. Let $X$, $Y$, and $Z$ be (smooth) varieties of the same dimension. Suppose that $X$, $Y$, and $Z$ can be embedded as codimension-one subvarieties of a common (smooth) variety $W$, such that

   • $X$ and $Y+Z$ are linearly equivalent as divisors in $W$, and
   • $Y$ and $Z$ intersect transversely in a disjoint union of (smooth) varieties $V_1, \dotsc, V_{\ell}$.

   Then $$\mathcal{A}(X) = \mathcal{A}(Y) + \mathcal{A}(Z) - \sum_i \mathcal{A}(V_i).$$

This assignment takes $X$ to its "Hirzebruch arithmetic number" $\mathcal{A}(X) = \chi(X, \mathcal{O}_X)$.

First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either

a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomolgically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.

Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.

Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.

Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.

Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.
In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.

First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either

a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomologically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.

Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.

Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.

Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.

Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.
In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.

Edit (added by Charles with Georges's permission): Fulton's axiomatic characterization may be described as follows: There is a unique assignment of a number $\mathcal{A}(X)$ to every [smooth irreducible projective variety over a fixed algebraically closed field] (hereafter simply "variety"), such that the following three axioms are satisfied:

1. $\mathcal{A}$ respects isomorphism classes.

2. If $X$ is a point, then $\mathcal{A}(X) = 1$.

3. Let $X$, $Y$, and $Z$ be (smooth) varieties of the same dimension. Suppose that $X$, $Y$, and $Z$ can be embedded as codimension-one subvarieties of a common (smooth) variety $W$, such that

   • $X$ and $Y+Z$ are linearly equivalent as divisors in $W$, and
   • $Y$ and $Z$ intersect transversely in a disjoint union of (smooth) varieties $V_1, \dotsc, V_{\ell}$.

   Then $$\mathcal{A}(X) = \mathcal{A}(Y) + \mathcal{A}(Z) - \sum_i \mathcal{A}(V_i).$$

This assignment takes $X$ to its "Hirzebruch arithmetic number" $\mathcal{A}(X) = \chi(X, \mathcal{O}_X)$.

First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either

a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomolgically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.

Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.

Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.

Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.

Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.

First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either

a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomolgically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.

Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.

Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.

Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.

Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.
In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.