Relative effective Cartier divisor

Let L be a line bundle on X and s a section of it such that $s:\; \backslash mathcal\{O\}\_X\; \backslash hookrightarrow\; L$ (in other words, s is a $\backslash mathcal\{O\}\_X(U)$-regular element for any open subset U.)

Choose some open cover $\backslash \{\; U\_i\; \backslash \}$ of X such that $L|\_\{U\_i\}\; \backslash simeq\; \backslash mathcal\{O\}\_X|\_\{U\_i\}$. For each i, through the isomorphisms, the restriction $s|\_\{U\_i\}$ corresponds to a nonzerodivisor $f\_i$ of $\backslash mathcal\{O\}\_X(U\_i)$. Now, define the closed subscheme $\backslash \{\; s\; =\; 0\; \backslash \}$ of X (called the zero-locus of the section s) by

:$\backslash \{\; s\; =\; 0\; \backslash \}\; \backslash cap\; U\_i\; =\; \backslash \{\; f\_i\; =\; 0\; \backslash \},$

where the right-hand side means the closed subscheme of $U\_i$ given by the ideal sheaf generated by $f\_i$. This is well-defined (i.e., they agree on the overlaps) since $f\_i/f\_j|\_\{U\_i\; \backslash cap\; U\_j\}$ is a unit element. For the same reason, the closed subscheme $\backslash \{\; s\; =\; 0\; \backslash \}$ is independent of the choice of local trivializations.

Equivalently, the zero locus of s can be constructed as a fiber of a morphism; namely, viewing L as the total space of it, the section s is a X-morphism of L: a morphism $s:\; X\; \backslash to\; L$ such that s followed by $L\; \backslash to\; X$ is the identity. Then $\backslash \{\; s\; =\; 0\; \backslash \}$ may be constructed as the fiber product of s and the zero-section embedding $s\_0:\; X\; \backslash to\; L$.

Finally, when $\backslash \{\; s\; =\; 0\; \backslash \}$ is flat over the base scheme S, it is an effective Cartier divisor on X over S. Furthermore, this construction exhausts all effective Cartier divisors on X as follows. Let D be an effective Cartier divisor and $I(D)$ denote the ideal sheaf of D. Because of locally-freeness, taking $I(D)^\{-1\}\; \backslash otimes\_\{\backslash mathcal\{O\}\_X\}\; -$ of $0\; \backslash to\; I(D)\; \backslash to\; \backslash mathcal\{O\}\_X\; \backslash to\; \backslash mathcal\{O\}\_D\; \backslash to\; 0$ gives the exact sequence

:$0\; \backslash to\; \backslash mathcal\{O\}\_X\; \backslash to\; I(D)^\{-1\}\; \backslash to\; I(D)^\{-1\}\; \backslash otimes\; \backslash mathcal\{O\}\_D\; \backslash to\; 0$

In particular, 1 in $\backslash Gamma(X,\; \backslash mathcal\{O\}\_X)$ can be identified with a section in $\backslash Gamma(X,\; I(D)^\{-1\})$, which we denote by $s\_D$.

Now we can repeat the early argument with $L\; =\; I(D)^\{-1\}$. Since D is an effective Cartier divisor, D is locally of the form $\backslash \{\; f\; =\; 0\; \backslash \}$ on $U\; =\; \backslash operatorname\{Spec\}(A)$ for some nonzerodivisor f in A. The trivialization $L|\_U\; =\; Af^\{-1\}\; \backslash overset\{\backslash sim\}\backslash to\; A$ is given by multiplication by f; in particular, 1 corresponds to f. Hence, the zero-locus of $s\_D$ is D.

• If D and D' are effective Cartier divisors, then the sum $D\; +\; D\text{'}$ is the effective Cartier divisor defined locally as $fg\; =\; 0$ if f, g give local equations for D and D' .
• If D is an effective Cartier divisor and $R\; \backslash to\; R\text{'}$ is a ring homomorphism, then $D\; \backslash times\_R\; R\text{'}$ is an effective Cartier divisor in $X\; \backslash times\_R\; R\text{'}$.
• If D is an effective Cartier divisor and $f:\; X\text{'}\; \backslash to\; X$ a flat morphism over R, then $D\text{'}\; =\; D\; \backslash times\_X\; X\text{'}$ is an effective Cartier divisor in X' with the ideal sheaf $I(D\text{'})\; =\; f^*\; (I(D))$.

From now on suppose X is a smooth curve (still over R). Let D be an effective Cartier divisor in X and assume it is proper over R (which is immediate if X is proper.) Then $\backslash Gamma(D,\; \backslash mathcal\{O\}\_D)$ is a locally free R-module of finite rank. This rank is called the degree of D and is denoted by $\backslash deg\; D$. It is a locally constant function on $\backslash operatorname\{Spec\}\; R$. If D and D' are proper effective Cartier divisors, then $D\; +\; D\text{'}$ is proper over R and $\backslash deg(D\; +\; D\text{'})\; =\; \backslash deg(D)\; +\; \backslash deg(D\text{'})$. Let $f:\; X\text{'}\; \backslash to\; X$ be a finite flat morphism. Then $\backslash deg(f^*\; D)\; =\; \backslash deg(f)\; \backslash deg(D)$.{{harvnb|Katz|Mazur|1985|loc=Lemma 1.2.8.}} On the other hand, a base change does not change degree: $\backslash deg(D\; \backslash times\_R\; R\text{'})\; =\; \backslash deg(D)$.{{harvnb|Katz|Mazur|1985|loc=Lemma 1.2.9.}}

A closed subscheme D of X is finite, flat and of finite presentation if and only if it is an effective Cartier divisor that is proper over R.{{harvnb|Katz|Mazur|1985|loc=Lemma 1.2.3.}}

Given an effective Cartier divisor D, there are two equivalent ways to associate Weil divisor $[D]$ to it.

{{reflist}}

• {{cite book

| last1 = Katz

| first1 = Nicholas M

| authorlink1 = Nick Katz

| last2=Mazur

|first2=Barry

|authorlink2=Barry Mazur

| title = Arithmetic Moduli of Elliptic Curves

| publisher = Princeton University Press

| year = 1985

| location =

| pages =

| url =

| doi =

| id =

| isbn =0-691-08352-5 }}

Category:Algebraic geometry