Many people have written crossovers of people both being a demigod and a wizard. However, those stories are INVALID. They are impossible. Why? I'll give you the short answer: All wizards are half-bloods. But not all half-bloods are wizards. Confused? Good. Then read this.
About four thousand years ago, somewhere in Ancient Greece, Hecate was in love. If you don't remember, Hecate is the goddess of magic and crossroads. She had her eye on a man named Aindrea. She eventually wooed him, and they had a daughter named Wand. Hecate was happy being with Aindrea, and even considered making him her immortal husband. That was before she met another man named Bob. Hecate, once seeing Bob, fell in love with him and got sick of Aindrea. And dumped him. Two years later, Hecate and Bob had a son named Wizard. How precious.
The daughter of Aindrea (Wand) was obsessed with sticks. Yes, everyone likes picking up sticks and poking bears with them, but Wand liked to pretend she was a magical goddess. She imagined being able to do almost anything she imagined just by waving a stick. She even made up a few phrases to say to perform a spell.
One day, Wand tried to do one of her spells: Wingardium Leviosa. However, instead of just fantasizing about it, she was set on doing it for real. Wand had inherited some power from Hecate, and using the magic she was able to make a piece of straw levitate, using her favorite stick. Eventually, she was able to do several kinds of spells, and got better and better at it.
Eventually, Wand became friends with the son of Bob, Wizard. Wizard was shown the spells that Wand could do, and he himself learned them. Soon, they got married. Yes, it's gross that they are half-siblings and they are still married, but this is history.
The descendants of Wand and Wizard grew, and they called themselves wizards. They called the magical sticks wands. Over the next 3800 years, many discoveries were made. Someone who's mom was a wizard and dad was Hades, bred the first hippogriff by bewitching an eagle to fall in love with a horse. Unicorns were bred by a witch who's mom was Aphrodite; by breeding a pegasus with a rhinoceros. The result wasn't a unicorn, but a very dangerous and large pegasus with a sharp horn. The descendants had to be bred with horses, until finally the unicorn lost all of its rhino aspects except for the horn.
Over the years, many Greek gods married wizards and had children which were even more extraordinary. Wizard-children of Demeter, for example, made most of the magical plants and herbs you see in Harry Potter.
When the gods decided to move to America around the 1800's, the wizards stayed. They had no trouble with taxes or freedom, so there was no reason to go to a completely new land. Wizards are still very scarce in America, so scarce that there isn't even a wizardry government there.
Wizards aren't just wizards. They are descendants of Hecate. This is why you can't be a half-blood and a wizard, because they are basically the same.
And this, is where the story begins.